You're not his mother and you're not his maid. Its entirely justified that you expect your partner to contribute equally around the house. Thats basically all that needs saying.

From my perspective you are making a mistake by focusing on his gaming and implying that his "unwinding time" is a problem in itself. He will use that against you in any argument. The fact is that he isn't holding up his end of household responsibilities, and the reasons for that are largely irrelevant. Stay focused on the crux of the matter.