But what if that breaks down? Better have a fabricator fabricator fabricator.
A sufficiently advanced fabricator ought to be able to replicate itself.
Though this is often forbidden for technobabble reasons as a plot point…
Fabricators are probably different to replicators. A fabricator makes things from raw materials, so you have to shovel dirt or something into it. A replicator (as per Star Trek) makes stuff out of thin air (and energy, presumably). But in theory a sufficiently advanced fabricator could manufacture another fabricator, so the first thing you’d do is make a spare.
Under the right conditions a Quaking Aspen can be quite successful.
The idea of putting a life form like that, in ideal conditions, on a alien world? That might be a creative vein I need to mine.
John Von Neumann’s famous mathematical analysis of self-reproducing machines showed that you don’t need an infinite regression of fabricators. That with the right set up a machine could copy itself any number of times and so could the copies ad infinitum.

Horses are an expensive commodity - they require a lot of land be dedicated to growing their food, and a lot of hands-on care - why would colonists waste any effort on them when they will have machines that can do hauling and carrying? And you have a few hundred colonists - can one of them really be spared to be a farrier, even part-time?
Machines also require a lot of effort, and specialized workers. They need to be built, out of parts that need to be manufactured, out of materials that needed to be mined and refined, and every step of that process requires specialized infrastructure - factories, mines, refineries, and the like. And everything in that supply chain needs routine maintenance, and constant supplies of fuel and lubrication.
Horses already self-replicate, using foodstuffs that we’d already be farming for our own uses. If they break down, they can heal, or be used as food and other resources.
If you have a small colony, horses are far more efficient. You’d bring machines to use for the first few years to establish the colony, with the expectation that horses would gradually replace the machines as the machines wear out and get broken up for parts.
The one advantage a machine has is that it can be parked for a week or a month, without worrying that it will die on you. But even that isn’t unlimited. Leave it sitting for too long, and things start to break. Seals dry out, steel rusts, lubricants evaporate, and soon it won’t start without a complete refurbishment.
Yeast and some grain seeds so you can make beer. Civilization started when people began to stay in one place, rather than roam around, they were waiting for the grain harvest so they could make beer. Egyptian pyramids were built by workers fed beer.
Horses are one of the stupidest domestic animals. If a horse sticks a foot into a fence the horse’s solution is to tear down the fence with its leg and kill itself rather than just pulling its foot ot of the fence. Anything goes wrong with a horse’s foot or leg so it goes lame sentences that horse to death. Just because horses worked well here in the past does not make having them on your new planet a good idea.
Oh, and garlic. You must bring some garlic or else why try to live at all.

Horses already self-replicate, using foodstuffs that we’d already be farming for our own uses. If they break down, they can heal, or be used as food and other resources.
Or to put it another way, a horse is already a horse fabricator (and a horse fabricator fabricator, and so on).

Horses are one of the stupidest domestic animals. If a horse sticks a foot into a fence the horse’s solution is to tear down the fence with its leg and kill itself rather than just pulling its foot ot of the fence. Anything goes wrong with a horse’s foot or leg so it goes lame sentences that horse to death. Just because horses worked well here in the past does not make having them on your new planet a good idea.
We can debate animals vs machines in general, but in horses vs other animals, horses have already been conclusively proven to be the best, by the people who actually used draft animals in real life. Given the proper equipment, horses are way better than anything else we’ve used. That’s why they eventually supplanted oxen and other animals pretty much everywhere that had the right combination of horses and equipment.
Those people weren’t stupid, and they didn’t have anywhere close to our levels of surplus wealth, such that they would waste so much time and effort on horses, if that investment didn’t pay off.
An alien planet with its own biosphere is likely to be a Biohazard Level 4 zone. Maybe not unadaptable to but likely not quickly or easily.
The problem is that we can only suppose what interstellar colonization would be like if we could by magic start today the way we are, just handed the transportation. By the time we can actually journey to other star systems our circumstances may be so different that all our presumptions today will be obsolete.

We can debate animals vs machines in general, but in horses vs other animals, horses have already been conclusively proven to be the best, by the people who actually used draft animals in real life.
I don’t think that’s true. I think they are the best animal for war. But i think people who weren’t already raising horses for war mostly used oxen and asses and dogs (and sometimes even sheep) to haul stuff. And oxen and asses to plow fields.

An alien planet with its own biosphere is likely to be a Biohazard Level 4 zone.
No, it’s likely to be, for practical purposes, a desert. Biohazard implies that their germs can infect us, but that would only be the case if our biochemistries are compatible, which they probably wouldn’t be.

And oxen and asses to plow fields.
Were you perhaps thinking of mules (donkey-horse hybrids)? Donkeys/asses are smaller than and not as strong as horses. Oxen were used for plowing for a very long time but the invention of the horse collar meant that horses became a lot more practical to use as draft animals. Oxen continued to be used when pulling very heavy loads but horses and mules came to predominate plowing. Also, horses tended to be bred and trained for their role; war horses especially were bred for temperament, and a horse raised to be ridden often couldn’t be used as a draft animal or vice-versa.
Like so many of these questions, it depends entirely upon the details. As discussed above, to do the classic colonization scenario, you either need a robust pipeline back to Earth, truly magical technology, or a ridiculously compatible biosphere. All of these things are doubtful. Currently, we can’t even imagine colonizing a place like Antarctica, much less the moon or mars.
It’s not impossible; it’s just insanely difficult. Even getting to an 18th century level of development would require massive amounts of effort, and is completely at odds with a brave homesteader mentality. You’d have to have people willing to work in primitive conditions for generations under authoritarian leaders to even begin. The alternative, libertarian approach leaves you with small agrarian enclaves and craftwork or home based industry that will take centuries to build up a population capable of supporting anything remotely resembling modern technology, even assuming the knowledge has been preserved and is available.

or a ridiculously compatible biosphere.
Indeed a lot of science fiction works on this theme postulate that it’s an alternate-timeline Earth or the distant past that’s being settled, because that’s the one setting guaranteed to be almost* 100% compatible with humans.
*the Mesozoic is a crapshoot. Almost no angiosperms until the middle Cretaceous, only primitive grasses, rather challenging megafauna, and the distinct possibility of diseases we’d have no adaptation to.
The Bible has a specific prohibition against harnessing and and an ass together. I have to think they were using oxen and donkeys (not mules).
It’s true that better horse collars made it practical to plow fields with horses, but there were still a lot of oxen used in colonial America. And no, they weren’t harnessing their milk cow. I have to think that oxen were still cheaper to use.

To make it easier, let’s assume a pre-human earth, with similar flora and fauna but no strange diseases, and the ship lands in a mild climate with forests.
This is an Eden fantasy without the religious part. A new world viable without humans. It has been done in SF stories over and over again.
Heinlein based many stories on this.

Machines also require a lot of effort, and specialized workers.
Sure. But machines are going to be needed whether you have the horses or not, so that effort is non-negotiable. Or were the colonists going to be mining, refining, generating electricity, making concrete, producing other materials etc. without machinery?
Or are y’all picturing some kind of Swiss Family Robinson affair where everything is made of bamboo and wood?

using foodstuffs that we’d already be farming for our own uses.
If you’re feeding horses with them, that’s so much less for humans to use.

they can heal
Aren’t horses the textbook example of not recovering easily from broken limbs.

be used as food and other resources
So would native animals, without the need to house and feed them first.

An alien planet with its own biosphere is likely to be a Biohazard Level 4 zone.
Once again - specifically excluded from the hypothetical by the OP.

a pre-human earth, with similar flora and fauna but no strange diseases
On a generation ship, where real estate is at a premium, every square inch counts—livestock are bulky, require room to graze, slurp up gallons of water, and produce copious amounts of poop. Insects, on the other hand, are efficient space-faring protein gurus: nutritionally, many bugs are over 50% protein by dry weight, loaded with essential amino acids, vitamins, and minerals. And they thrive in tidy, stackable trays, sip minimal water, and can even be fed human waste.
The only hurdle? The ick factor. Folks would have to get on board with such delicacies as Bug Bourguignon, Cockroach au Vin, and Mealworm à l’Orange. Garçon, my order please!:
When we’re floating light-years from the nearest McDonalds, our palates must adapt! If you’re feeling sentimental, you could still pack a few livestock embryos for special occasions—future cosmic cows. At least our great, great, great grandkids will eat well.