Playing non-4K games on a 4K TV?

Modern TVs have Gamer-specific settings which switch most if not all post-processing off.
Depending on the model there may even be Variable Refresh Rate for example or other stuff for modern consoles.
Best to search which TV-Model you're looking to get and read or watch reviews.
Edit: I can recommmend the british channel HDTVtest for no-nonsense reviews and guides.
https://www.youtube.com/@hdtvtest

rtings.com has great reviews including recommended settings and input lag measurements.
I'm guessing the situation is pretty similar for your old TV too though, all TVs have some level of input lag, nothing special about 4k in that regard. You can still run games at 1080p on the 4k monitor if your rig can't keep up.
IMHO you wan to turn off all that post-processing stuff anyway (including for TV/Movies).

My experience,

Newer Sony Bravia TVs seem to have really nice 4K upscaling from lower resolutions, and decent input lag in game mode. I have a 65" Sony Bravia from ~2019, I think it's a X900F. Nintendo Switch games look like they're running native 4K.

I also have a 55" LG CX OLED. This has better input lag in game mode, but the upscaling from 1080p is considerably less impressive than the 3 year older Bravia

4K is exactly 4 times 1080p so scaling is very easy for the TV, you can run any game at 1080p if the game do not support it or the graphic card is not good enough to run it at 4K at a decent frame rate.

If you're looking for high-end TVs, watch the HDTVTest-channel on YT: Vincent Teoh is the guru of high-end TVs.

Short and simple answer: You can run all your games on a 4K-TV. They get upscaled.
The nitty-gritty is where it get's a little more complicated. We have to get into a more detailed conversation to figure out your needs and budget. Or I can simply recommend the best gaming 4K-TV to you.

Let me give you a brief review of how I am doing it:

• Very high-end 4K-TV with 4 HDMI 2.1-ports (sadly not full bandwith, but 40GBpS)
• Xbox Series X straight into one of the HDMI ports for native 4K 120FPS gaming
• PS5 straight into another one of the HDMI ports for native 4K 60FPS gaming
• Steam-Link (Box) connected to the AVR which is plugged into the eArc-HDMI-port of the TV

Sadly, the Steam-Link Box can only do Full-HD-gaming (so 1080p), but that's okay, because my PC also only can do Full-HD-gaming. The Full-HD games get upscaled via TV to 4K, so I can play every Steam game on the TV.
BUT they look a little bit worse: colours are a little washed out, they are a little darker, and you have slightly more input lag. The same games look more vibrant if I play the native 4K-version on one of the native 4K-consoles. But the effects are really minimal, so that I do enjoy my upscaled PC games on the TV in my living room.

That's still a basic answer by me, by the way :D It does get more complicated, if you want to be more detailed.

Better to get a "future proof" 4k then a 1080p now. What if you want to watch a 4k movie? Also as others have said, it can do 1080p fine. It will either upscale (if its pricier) or simple change the resolution down to 1080p. The higher res screen will still look better even if you won't get more details, etc. If you ever get a steam-deck, I believe it can do 4k upscaling.

Upscaling to 4K isn't a problem so long as you don't care about it being pixel-perfect (such as for pixel art and retro games), as that requires integer scaling and most TV manufacturers don't include it. 4K is a perfect multiple of 1080p with exactly double the vertical and horizontal pixels, so if you can find a 4K TV with integer scaling 1080p signals will look extremely crisp, but in general TVs will instead only be able to upscale in a blurry fashion.