I'm big on thanking so don't have a problem with that at all. Newscasters say 'thank you ' to their colleagues when they mean 'ok, I'm going to take over now'. I use it in meetings all the time to move things alone
Your issue as I see it, is more about responsibilities, in that you appear to be responsible for , well, everything. He doesn't need to be 'Santa's little helper' or even your boss. What you need is each to have areas of responsibility.
Divide up what there is to do however suits you both. Just accept that it needs to be divided, that not everything yours.
For example, cooking could be divided into different days, but who makes sure the ingredients are available / plans the meals : what to buy? Alternate days cooking / washing up? Find a way that takes up roughly the same amount of time over all the things needing doing.
Him doing the gardening (sitting in the shed planning throughout winter and spring ) and you shopping/cooking/washing up, child wrangling and buying their clothes is not a good split.
I'm not suggesting it's easy, but if is necessary. You can adjust as you go along just so long as you both feel it's fair. Ideally you both genuinely feel the other is doing more, not that you do more and he accepts that you do more.
It's one of had most difficult things to do at work. And at home. It's definitely worth spending a bit of time on it to get it right.