I think the grass is always greener.
I’ve been married with a baby, and single with a young child. And though I’ve not been a SAHM I’ve done 13 months maternity leave with all the childcare sorting.
Yes, he doesn’t have the monotony of full time child caring.
But he either has a job that isn’t demanding - in which case he has his share of demanding - or his job is demanding, in which case you’re luckily missing out on the work shit that he’s getting. Preparing for pointless annual appraisals, anyone?!
As for appreciating the housework... should he really say “thanks for doing all the cleaning?”. Well, only if you want to say “thanks for all the money earning”. Of course, he should then say “thanks for enabling the money earning”... etc.
You probably both need to talk about how you show appreciation.
You also need to not be a martyr. If you don’t want to deal with all the nursery choosing - you should have told him. There is no reason him not to google options, take a day off t have several visits. Those things are not onerous.
Mental load is real, but in my opinion it’s become somewhat overhyped, and makes people just feel more discontent. Now we have a label to complain that we have so much mental load. Well, no - mostly what you have is just living life. You say he doesn’t have that - but you don’t have the work demands that he has. I have far more mental load keeping on of work than running my home and managing my child’s day.
Make sure he’s pulling his weight.
Don’t opt him out of things (like nursery?) if you’re not happy to do it alone.
Find a way to communicate your appreciation both ways (I don’t give a shit if my husband notices that I’ve washed up - I feel loved and appreciated from his other actions).
And don’t get hung up on mental load - because he probably does have more than you think he has, and because some of the mental load is anyway just life.