Husband just goes and does his own thing

Yes, it does sound like you need to communicate that you'd like to spend more of his very limited time at home doing something together. It's reasonable to want to spend time together to reconnect, but he may not think of watching TV together as quality time, so it may not occur to him to ask if you'd like to join him. Or if you're already doing something, he may assume you're busy, then just go do his own thing.

Are you happy watching what he's watching or will it turn into a whole 'thing' with the two of you discussing what to watch, snacks, etc? If it's more of a production, I can see why he might want to just watch what he finds interesting at the moment, sometimes, but it's also completely normal that you want to spend time together when you can.

Yes I agree. It’s starting his own past time independently, rather than suggesting a shared one. If moments together are rare then yes I’d be put out to have to join him when he’s already (at a guess) 45 minutes in to a film. You’re in a situation where you can’t really join in because you’ve missed half of it, so you’re just waiting for him to finish the film to do something shared, but then the evenings gone. That’s annoying

I think the issue here is more that he’s sat down to watch a film. So you’re either going to miss the start of it or not watch it at all. Would be better if he had sat and watched general TV or scrolled his phone, then chosen and watched a film together, especially if you don’t get to spend much time together. Sounds like he’s got used to doing his own thing.

Seems like as
yoi don’t spend much time together he is used to just doing things on his own and doesn’t think about it. Have you told him this upsets you?

I don't read this as an issue of task division.

It is about him starting his own past time activity without suggesting/offering to do something together. Even if it was looking at the very same movie together (working to be done at a specific time/waiting for her if on demand TV).

If course they don't have to do everything together all the time, but it isn't unreasonable to want to spend some time together and it seems they don't have many opportunities.

Im not sure what you are asking but basically it seems as if you're expecting him to read your mind.
He's just relaxing in his own home. Why don't you try communicating with him and telling him what it is you're after.
This sounds a bit mad tbh.

Is there a back story? It doesn’t sound too bad tbh, more like you both were doing some jobs - he’s done the kitchen and you did other things - and when he finished he watched TV. Was he supposed to do more tasks or do you feel that it’s not a fair share of the load? Maybe worth having a calm conversation about this because it might not be obvious to him that you’re unhappy with how things are shared?

Don't know whether it is 'cold', just the way he does things and seems like that makes you feel rejected/ sidelined/unimportant/taken for granted (not quite sure whether these are the right thing).

What does he say when you discuss it? (Ideally calmly, using 'i feel' structure)

Perhaps you can mention doing something together in the morning so it's not left to chance in the evening?

Although tbh, I really don't think there's anything wrong with what he's done here.