No.

There are NO reasonably priced projectors with HDMI 2.1 on them and 4K/120hz support.

It would be fair to read about the reality of game consoles and their actual ability to deliver that resolution and frame rate anyway.

But, a TV is a FAR better value for this type of need. I'm not well versed on cheaper options, but the 55 inch LG C2 OLED is literally $1,000 which would be what a fairly decent 1080p projector would cost.

For even less, the Hisense 55U8H has a 55" model for $650. That's the way I'd go if my kid were really wanting 4K/120hz support.
https://www.amazon.com/gp/product/B0B5YZNZTZ?tag=rtings-tv-r-amazon-20&ie=UTF8

As it is, my kid is much more casual about his gaming on a big screen and he was perfectly happy this last year (his first in college) with a 50" Amazon FireTV which I nabbed on Prime Day for $100.
https://www.amazon.com/gp/product/B08SVZ775L/ref=ppx_yo_dt_b_asin_title_o00_s00?ie=UTF8&psc=1