My (29F) husband (29M) 'rests his eyes' every night but doesn't do anything about it

Wasn't sure to put this in parenting or daddit, or what have you.
We've been married 5 years, together 12. Have a 3.5 year old. I work in health care, he recently (3 m ago) got a new managers position after vying for it for years. It's tough for sure, getting a new position right before COVID but he's been doing a great job. I've been taking time off work thankfully to juggle kiddo.
Ever since we had kiddo (I had major PPD, we went to counseling to work thru things after baby) he is always. tired. We both got put on antidepressants, and separately went to therapy to work thru issues (home, work, kiddo, etc) and were doing well.
He works a general 30~40 work week. He answers emails and calls at home and weekends. Supper is either made when he comes in, or we are fortunate enough to order in. The only "chores" I ask him to do is mow the lawn as our lawnmower is old and I can't pull the cord well enough to start it. (We need a new one with push button.)
He comes home, eats, plays with kiddo before he bedtime at 6pm or 7pm (if she naps) then he puts her to bed. But after supper, he usually sticks her with the TV as I clean up and he 'rests his eyes' on the couch and snores. She's calling for him asking to play, "what's this" or whatever. I get frustrated asking him to not, but he says it's like meditating he's adamant it's not sleeping.
After she goes to bed, he stays up till about 11pm gaming, then goes to bed. Up at about 6am then off to work at 7am for work. then complains and slugs about all morning. Drinks 2 pots of coffee and stops by Tims before work too, and 2 cups of coffee at night, too.
What can I do? Am I wrong? I'm not a martyr but I feel like he could do more.
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TL;DR: Husband 'sleep'/but ITS NOT SLEEPING!! (he says) on the couch nightly but doesn't do anything about it/games all night, then complains every morning. What can I do?! How can I help.


Source.