Don't need to get too complicated. You're asking enthusiasts what you should buy, and they will suggest enthusiast-grade products like DACs and dedicated mics. You don't need any of that.
Listen to people who suggest a quality headset (most of the time are not typical gaming headset companies) that don't make your wallet cry.
I've used the Sennheiser PC38X for years now. It's an open-back headset, very comfortable, legit headphone quality sound, built-in mic, for like $189 (used to be $139 sadly it went up). Many enthusiasts suggest it.
By far the best, most comfortable gaming headset I’ve owned is the PC38X from Drop. It’s a Sennheiser headset.
Lightweight, fantastic soundstage, open back.
Shits all over other brands 2-3 times its price and I’ve had many.
Drop/Sennheiser PC38X Gaming Headset. I love 'em. These are open back. I don't know if Sennheiser makes a closed back version anymore.
You get Sennheiser sound in a headset. Mic mutes when you flip it up. No software.
A price range is pretty important for these types of questions. Also, do you want wireless only? With PS5 Pro, I’ve personally used Drop/ Sennheiser PC38x (predecessor PC37x would be cheaper option) for multiplayer or fast paced games (like Returnal) for good directional audio and HiFiMan Edition XS for cinematic single player games. You would need to plug them into your controller in the 3.5mm jack. I’m not into the overpriced and underperforming “gamer brand” headsets. A decent Sennheiser headset typically blows them into the weeds and lasts years longer.
I have higher end headsets that would blow your mind like the HD490 Pro and especially the HE1000se I use on PC. Those are probably going to be way too pricey for this discussion.
I'd probably recommend Drop/ Sennheiser PC38x for multiplayer/ directional audio and HiFiMan Edition XS (or perhaps Sundara if you have small ears; not comfortable for me) for cinematic single player games for fairly affordable headsets within a few hundred bucks. I've read good things about the Audeze Maxwell. Going much higher will probably require an amp and get MUCH pricier.
For PC and listening to music, I use an HE1000SE, which will challenge your statement about budget not mattering. It's by far the best set of headphones I've ever used for gaming (including the also excellent Sennheiser HD 490 Pro, which I also own) or music but definitely not low budget.
The Sennheisers I have are the PC38x and HD490 Pro. Both are good for their respective price ranges overall. I have minor comfort issues with them because I have a large head and large ears. Have to use the cloth pads for the HD490 Pro. Sound for gaming is great, though.
Top Gaming Headsets on Reddit
1
SteelSeries
Arctis Nova 7 Series
83% positive of 301 users
Multi-device audio & comfort king, but buggy software.
4K4llDay
