Firstly forgive the length of this, I'm crap at being succinct!
Kindly, it sounds like you're in the trenches right now. Without knowing you it doesn't sound a million miles off from my experience of life a few years ago.
Working ft with a young family, shouldering the financial burden whilst I was on mat leave etc was really difficult for DH. Life felt hard for me too but in totally different ways. He worked hard in a demanding job, parented hard, did his share of the household stuff, but there was a limit to his capacity. He seemed to crave time where he could totally shirk the feeling of responsibility and have some hours which resembled his old life. He genuinely retreated into himself, like a coping mechanism. You say watching football or listening to a podcast, in this house it was heading out for a run as soon as the kids were down, taking on the tasks of weekly shops and batch cooking of an evening where he'd leave the house or be in the kitchen solo, or staying up till the wee hours playing playstation.
At the time it was difficult to swallow (that there didn't seem to be capacity for us to spend time together) and I worried for our relationship, but now the children are a bit older it's much much better. It's taken quite a bit of communication, mostly driven by me. I do understand how frustrating it can be to be the person asking/demanding but in my opinion it was worth it for the greater good long term. I'm better at that stuff than he is, and as a rule we try to play to our strengths for the good of the unit.
I'm quite an organised planner type, he's more of a wing it/go with the flow type. We each have our moments where these traits are useful! Eventually I looked at the week and scheduled in time that we'd each have to ourselves (for exercising, socialising, ignoring the world and playing computer games....whatever you like) plus a date night as often as we can. Mostly these were at home (a games night for example) as we didn't have easy access to babysitters plus could rarely afford to both spend on going out and pay for childcare, but occasionally as you say we prioritised leaving the house for a meal (brunches were often cheaper than dinners and it was easier to go out whilst the kids were at nursery or arrange a playdate than find and pay for an evening babysitter). For example on Friday he's taken the morning off and we're going to an early matinee at the cinema, whilst the kids are at school, before the summer holiday madness kicks in. We also swapped a few chores around to make things more balanced considering I was working PT and him FT.
I know he functions better if he exercises regularly, the same as I need time with people outside of the house, and once a year we each get a full weekend off to go and do something with our mates or solo, whatever you like. All this helps us remember who we were before we had the children, and it has improved the feeling of being a team tremendously.
Idk if this is helpful, but if I could tell previous me that all the chats/efforts etc would be worth it and in a few years time our house will be so full of laughter and we'll feel like a real team, that would have been reassuring.