A stay at home dad, even if he actually works at home, is usually supposed to take care of the kids and household chores. What if you really had no time to do these things? Would you hire someone to help?
You have to get them done somehow. If you put them off your house will become chaotic and everyone will suffer. It will be like the snowball effect. Either make time or hire someone.
I am a single dad and I don't have time to keep things clean. At one point I just let things go, and Jdrake is right, that doesn't turn out well. I hire a woman to come in two times a week and it makes a world of difference.
Hiring someone to come in and help out is not uncommon at all. It's not something just for the wealthy. They will even clean apartments. Some people just hate to clean or like you really are just too busy to worry about things like that. It's good to have help.
I'd say this all comes down to priorities. You can hire a helper but if you still don't prioritize cleaning the house then it will just become messy again after the helper has left.