mic not important as I don't speak or go online that often,
my environment is quiet as its mainly my dog that is sitting next to me as I thankfully have 2 living rooms one for me and the other for the wife
Then forget term head
set used to sell cheap production trinkets at overprice.
Anyway for that budget you could easily also get ModMic capable to turning any proper head
phone to headset.
For quiet environment open design headphone is better.
Besides isolating sound closed headphones also isolate heat more, making them cause sweating more easily, especially if with (p)leather pads.
And acoustically closed design is huge challenge/drag compared to open design.
With binaural recording or good binaural sound simulation (from 5.1 of the game) good open headphones can recreate very nice feeling of being in there with 360 directionality and even sense of distance to sound source.
Also in open design above neutral bass doesn't that easily start drowning detail under it.
Though for best "competitiveness"/details neutral bass is always advantage.
And for example AKG K702 costing mighty ~£110 are one of the absolutely best headphones for competitive gaming.
Its bass actually goes pretty low for open dynamic driver headphone, but it always takes back seat to details unless source signal says otherwise.
K712 is AKG's vision of that fun above neutral bass punch.
Otherwise it's pretty much equal, details just aren't as easily recognized from simultaneous low frequency sounds lowering competitiveness one notch.
Beyerdynamic DT990 has similar fun sound, with little more bass immersion and notch smaller soundstage for little lower competitiveness.
Compared to those capabilities of cheap closed design gaming trinket are like putting head into bucket under water.
If you absolutely hate bass then Audio-Technica ATH-AD700x would be otherwise at AKG K701/702 level.