Hi, some may say "It's just a game!" but I tend to get pretty darn immersed when I play so here's my issue:

I am thane of Whiterun and Haffingar. So I have housecarls...but I only ask them to help me in the most dire of situations because I feel like an (donkey)hole if I keep pulling them out of their home with me.

So: what do you think? Do housecarls like going out, or would they prefer to stay home?

They clean my house and make me food, they'd just get in my way if they would come out from the house.

I love m housecarls. They're like squires, ready for battle, holding my armor and weapons. All this viking-flavor is a plus

They are sworn to carry your burden, so....why not? If that "burden" happens to be death, so be it.

In regards to the main idea of the OP - whether or not Housecarls like to leave the home - I'd have to say that yes, they do. The Housecarls are warriors, sworn to defend your life and fight by your side... I imagine that for most of them it's an honor to take part in the Dragonborn's great deeds, and when I play I try to keep my Housecarl with me, and alive as a sort of perpetual sidekick.

In my opinion, the average Housecarl would feel a little humiliated if they were left at home to be a sort of glorified maid.

I agree with the O.P. ... generally the housecarls seem quite happy just sitting around the house, they never seem to be clamouring to be taken out. This would be annoying in a follower, who are usually fairly enthusiastic, or a mercenary, who you've paid for; but the 'carls are essentially just temped-out by the Jarls. Way I see it, demanding too much of them is just exploitation.
That being said, I wish you could just give them a gratuity and a handshake and send them on their way. I'd rather not have to deal with sub-contractors in my game.