Most of Batman's whole persona doesn't make much sense - being a billionaire who dresses up like a flying rodent is more than a little silly. But hanging out in a gross cave filled with bats all the time? That's INSANE. You're a billionaire! You own a mansion! Why are you chilling in a dank cave filled with disease-ridden bats instead of enjoying some champagne in your ultra-fancy house?
But it's even worse for Alfred - the only person Batman truly trusts, and thus the only one he can trust with his household chores, which include:
Really, the least Batman could do for his faithful manservant is not make HIM walk around a cave all the time too.