Personally I haven’t faced any weird heating issues on my Phone (3) in well over a year of use. My phone got warmer during the last few weeks, however it’s summer with frequent temperatures of 30-40°C in the sun and partly very poor cellular service, so it’s no surprise the phone got very warm in such situations. I guess trying to game on it wouldn’t have really been possible.
However…
I don’t play any mobile games. Like at all, so this doesn’t help you in any way. The last time I had some sort of mobile game installed on a phone was when Angry Birds got released. 😅
If gaming is a top priority for you and just by judging your post it really seems to be the case, honestly why not get a dedicated gaming phone in the first place? Something where you know it does have the cooling capabilities to handle even demanding games during summer.