Sorry if I'm dull, but for me it's a bit hard to decide what your actual problem is. One time the picture is too dark, but then, when "the blacks look grey", it's kind of "too bright" again, isn't it? Moreover, you did not say the problem is with games in general (which would indicate wrong RGB levels setting), but you explicitely said it is when using Game Mode (which would collapse things down to a problem with picture settings).
So. Is the problem present with games always, regardless of Scene Mode? Or are games okay with Scene General or Scene Cinema, and look only bad with Scene Game?
Simple test. Bring on a dark screen with lots of black. The TV's native black point is at "Brightness=50", exactly there and nowhere else. Check in a very dark room. When things are correct, then the plain-black level should not change with Brightness<<50, and full-black should immediately begin to brighten up with Brightness=51..52..53.
If you need to bring brightness down to ~36 for full-black to get truly black, then there is an input level mistake (input="limited", TV expecting "full").
If true-black only starts to brighten up from Brightness=64 upwards, then there is an input level mistake (input="full", TV expecting "limited").