I want to thank everyone for their thoughts and proposed solution. I regret we cannot physically relocate the source router to a central location (nor run an extended ethernet cable to better location) because of our house’s layout, the mesh network sounds like ideal solution to implement at some point in the future when money’s available. For now, we’re going with a $30 booster and see how that goes. Sorry, but money is the limiting factor for awhile. 
I appreciate the thought, but I must pass on your offer.
Those microwave repeaters use directional antennas with narrow beam transmission to avoid interfering with one another.
I failed to address this, and didn’t notice until the edit window passed. That’s more-or-less the approach taken by those expensive mesh WiFi network systems. They use a separate WiFi network to communicate and pass network traffic to one another and so don’t affect the “main” network with end-user devices on it. Similarly, you can deploy wired WiFi extenders or mesh networks that use a hard-wired network as a backhaul.
But for functional reasons as well as cost, your common WiFi repeater makes use of only one communication network for everything.
I don’t think it’s necessarily true repeaters halve the bandwidth, is it? They might roughly double the latency, because each one has to receive and re-transmit the packets they get, but that’s not the same thing as cutting the bandwidth from 100 Mb/sec to 50 Mb/sec (and I hope it’s megabits that you’re describing, not megabytes… because if you actually have 100 MB/sec wireless, that probably means you have gigabit ethernet and you need much more expensive hardware to get that around the house).
In any case, your average consumer router and repeater setup is probably good for ~300 Mb/s more when it has a good signal. If it’s 50 in some rooms, chances are good that the signal is too weak, and a repeater might help that even if it bumps the latency up by a few tens of milliseconds (like from 10ms to 30-40ms, no big deal unless you’re doing competitive gaming).
More practically, client devices can be finicky about which access point they want to connect to. If you move from room to room, you probably have to turn the wifi off and on to force a reconnection to the repeater instead of the central router (assuming both have the same SSID).
It depends on what you buy. The oldest simplest routers were simplex, one channel on one band. Most modern routers are dual or tri band, so you can use one channel in each band. More expensive routers are MU-MIMO, and can do multiple full-speed subchannels, including full duplex. Also there are some dual-channel units, but that’s specialist equipment.
They generally do as half the frequency bandwidth becomes dedicated to communicating with the router, leaving the other half for users.
Don’t think of it on the packet level but on the antenna level, i.e. not source encoding but channel encoding if you are familiar with digital communications. The pipe itself, i.e. amount of frequency for encoding, actually gets reduced in half, so the number of bits that can be pushed through is reduced in half.
That is, a single packet itself takes twice as long to get there, rather than being pushed twice at the original higher speed. There’s also a bit of processing delay as well with the retransmission, but that’s not typically too big.
Not a specific answer but. When purchasing a WIFI router, you might want to get one with antennas that screw in. This allows you to extend the antenna via a cable.
I have my router in the basement. I do not have to. But if I wanted to. I could get extension cables and mount the 3 antennas along the most favorable points in my basement. If your router is in the basement, you may also want to angle the antennas so they radiate best. An antenna radiates in a horizontal plane when it is vertical. Directly above and below are weak zones. So areas above and below will get weaker coverage. I have a 3 antenna router in my basement. I orient the unit parallel to the length of my house. It is about center of the length of my house. I tilt the outside antennas inward. So they radiate upward towards the ends of the house. I tilt the center one back, so it radiates upward to the center of the house. If I left them all vertical. The best coverage would be in the basement. Tilt them and you move the coverage up. Shorter antennas radiate at a wider angle in general. But any spot is weaker. A longer tuned antenna radiates a narrower horizontal beam. Great reception in that beam, but not out of it.
It should be noted that even the slower speed is still waaaay faster than the home internet is (probably).
50 MB/s = 400 mbps. I would be surprised if the family’s home internet speed is over 200 mbps. Solid 4K video streaming is 15-25 mbps. Much, much slower than the home’s internal network speeds.
Where this all matters is if you have a big family and everyone is trying to stream their own 4K video. It starts to add up. But mom, dad and daughter all doing that is, at most, needing 75 mbps internally…still way below the 400 mbps that is possible in the home.
Other special needs for fast speed might matter on the internal home network but for most people they will never know the difference because the bottleneck is the internet speed to their home which is much less than their internal network speed.
Overall speed is an average. Where you can run into trouble is in a dicey part of the home where the signal can vary a lot (and they can and do vary a lot). Averaged over time 50 MB/s seems good but it can happen where it cuts out completely then reconnects, goes fast then cuts out again.
I have a co-worker with this problem so Zoom meetings with him go from perfectly fine to him being frozen for 30 seconds then back. Very annoying.