Knowing the exact model would be useful, so find out if you can. Maybe we tend to fuss over PSUs a lot on this forum but unfortunately there's still too many unreliable ones.
I'm not worried but i would like to know.
I think if you stay with your current system you have three choices, or tiers:
$200-300 - gets you basicaly a GPU comparable to what you already own. A 4060 might be a modest improvement, but if you're spending money, get more than a modest improvement.
~$500 - at this price point i think the RX 7800XT is unbeatable. Sometimes you can find one on sale going for something like $450-480 or less if you're lucky.
$600-700 - this is 4070 Super or higher territory and is the most you can get imo.
https://pcpartpicker.com/product/Y3...tx-4070-super-12-gb-video-card-zt-d40720h-10m
Depends on the game but some games will want more than a 9600K. In any case, if you're considering spending the money a 4080 Super goes for, you'd be much better off buying a completely new PC.