Not at all. I live in Chicago and I’m used to noise 24/7. In Manchester, I usually stay on Portland Street and I’m well aware that this is a nightlife area and not the place where the only noise is birds chirping.
I live in suburban Chicago, and we’re on a street that, while not super-busy, does have some traffic all night long, including, sometimes, loud trucks. I sleep through that consistently.
However…there is now a martini bar across the street from us (it wasn’t there when we moved in). Every few weeks, we’ll have bar patrons who leave the bar (especially at closing time), who want to keep having a conversation – and, as they’ve just left a bar, most of them are at least a little inebriated, which leads to them talking louder than they realize. So, they stand outside the bar, under the awning (which is directly across the street from our bedroom window) – maybe it’s the angle, or the hard brick wall behind them, but the sounds of those conversations get piped directly into our bedroom, and wake us up nearly every time.
If they leave right away, it’s not too bad, but some of them linger for a long while, talking and laughing – after ten or fifteen minutes of that, that’s when I get up, put on shoes, walk across the street, and ask them to go home. 
I’ve stayed in city center hotels dozens of times and only a very few times was street traffic even an issue and it didn’t rise to a “problem” per the OP. Most of the time the room I get is too far away horizontally or vertically for it to even be an issue, and a lot of the time I am pleasantly surprised by the lack of loud traffic despite my room facing a medium or large street.
But I can think of a few other instances where outside noise other than traffic was a problem in city center hotels. In the French Quarter, they pipe the blues rock onto the streets at 7 am sharp no matter how late you stayed up the previous night.
And in Mayfair I had a room across the street from a pub where the first night was quiet but the second night there was probably a big football match on because there were periodic crowd roars until midnight, and the sun shining into my room until 11 or so didn’t help things either.
Growing up, my cousin stayed at our house one night; she couldn’t sleep as it was too quiet. When I stayed there, I couldn’t sleep because it was too noisy; they lived on a major 4-lane road while we were in a residential development.
As an adult, I only remember two times -
[ul]
[li]Cincinasty - beer bottle bowling league in the hotel hallway. Everytime I started to call the front desk/security, they’d stop for a while; I guess to go get another round.[/li][li]Midtown-Manhattan - was fine until 5am when the trashtruck showed up with it’s beep-beep backup alarm & he moved the truck for each of the multiple dumpsters.[/li][/ul]
Before they closed down, we sometimes stayed at Turtle Pier in St Martin. It was a quaint little hotel right on the water of Simpson Bay and just a short walk to Princess Julianna Airport.
You would think that a hotel by the airport would be noisy, but the only time we were at the hotel was to sleep and there were no night flights. During the day we were out and about.
No, it doesn’t bother me, but I am also used to city noise. Traffic noise generally blends into a constant hum that isn’t annoying. Other noises can very occasionally be annoying, but not traffic. I’ve stayed in a hotel in central Manchester and it was really quiet, so TBH it’s not something I’d necessarily expect to encounter there.
I guess the problem for some people on holiday is not just the noise, it’s that they don’t sleep well in a new place. So the noise is more of a problem in a hotel than it would be at home. My ears don’t work well with earplugs (so I’m glad noise doesn’t usually bother me), and I’m not the only one. Plus, if you weren’t expecting the noise to be that loud, or weren’t expecting it to bother you, then you’ll only really know that after you’ve gone to bed, when it’s difficult to purchase earplugs.
Not everyone goes on holiday regularly so they might not know that taking earplugs is often a good idea. Probably a good idea for hotels to sell them or even have them available for free. The only time I stayed in a hotel with a bad noise problem (a really excessively loud concert right outside my window - couldn’t hold a conversation) I went down and asked front desk if they had any earplugs, and they didn’t.
While staying in a hotel I’m much more bothered by noise inside the hotel than outside of it. I kind of like traffic noises and even use it as an option for my white noise machine. I don’t like people being noisy in the hallways in the middle of the night because they’re drunk or arguing.