Yes everyone moans to their friends sometimes. I might tell a friend if we have had a big argument about something. But I will usually give his side as well (as he is a reasonable person) and its rare.
Texting a mate about having to tumble dry something when it sounds a. Like he has made it up or at the least really exaggerated it and b. Has spent 5 hours down the pub, just seems really disrespectful and petty.
But then someone who is content with their partner doing the vast majority of their share of house and child work, and who fucks off to the pub for the majority of a day on the weekend, sounds like he already thinks everything domestic is your responsibility and has no respect for you as an equal. So all the text shows is it proves what his actions suggest he already thinks about you.
So given you already know he thinks this about you, nothing has changed, so I wouldn't confront him. I'd be reconsidering the relationship. It sounds like you want to work on him increasing his share of the load...but I'm not sure I'd want to continue a relationship where I had to force someone to effectively act like they pretend to respect me, and be in charge of getting them to do stuff like a child.
That's what your husbands message reminds me of actually, a teen messaging a mate that they hate their mum because she only washed and ironed and folded his clothes and he came back from a long day hanging out with his mates and she had the audacity to leave these clothes on his bed for him to put away when he could have been gaming. Ungrateful and completely not in the real world