I don't believe the Windows spatial codecs (so Dolby Atmos for headphones, DTS:X headphone, Windows Sonic) are supported in either the latest COD:MW3/Warzone 2 or Apex Legends - if those games were the ones you were testing with.
It may be using the Virtual Surround from the G6 is better in that scenario, i.e. just run that on it's own amongst the SBX enhancements. To be honest these days I tend to prefer plain stereo and then let either the in-game engine do the work, or use DTS:X on Xbox or Tempest 3D on PS5 - for games that support those although I don't really play any multiplayer/competitive games these days. My G6 isn't connected up atm.
The headphone question is more difficult in that there is subjectivity involved. There would be nothing wrong with starting with the HD 599s are seeing if you like open-backs, particularly if the budget is set. The HD 599 is long standing and well regarded open-back pair for gaming. Definitely been superseded but whether you need anything more is going to be preference.
I used to have the HD 598s as they were my first 'audiophile' headphones. They were great for gaming and a step up from some of the awful gaming headsets I had used previously (although headsets tend to be much better these days). The HD 560s is more neutral/clinical, more detail, a bit moire siblent, better imaging and more soundstage. But they will cost twice the price. The TYGRs are more v-shaped than the 560s but have very similar qualities. Indeed they are re-tuned DT 990s but with much smoother tuning, which makes the treble and bass more pleasant. I currently own both and love them for gaming, music and general listening on PC (much less on console given the backwards adoption of open audio standards). I have given up trying to choose between them and have kept both. In terms of the 560s vs TYGR if you like the sound tuning of the DT 770 and love the comfort and fit then the TYGR as another Beyer will be much closer in sound tuning and does have the exact same frame, similar pads etc.
What I would add, is that the DT 770s are solid for gaming (particularly amongst closed-backs but not their 'strength') but open-backs tend to have more soundstage and better imaging which can help with gaming. I think closed-backs tend to lean towards arena shooters or where concentration/immersion from outside noise is needed.
Sorry a bit of a long winded answer but hope that helps. There isn't a wrong answer per-say and it would be wise to look around for reviews, reddit opinions etc. as well.