A home is only as gross as the resident allows it to be. I have an indoor cat and guests to my home who don’t know it are uniformly astonished when she comes sauntering into the room, because there’s no obvious evidence of her. (Even her cat bed in the living room looks at casual glance like a piece of furniture.) But I like living in a clean home and make an effort to keep it that way. I vacuum, dust, mop, etc. regularly, and would do so if I didn’t live with a cat. Add in taking care of the litter box properly, and a cat living in my home is no problem. BTW, my cat is never, ever allowed on the kitchen counters or the dining room table.
Personally, I have been far more disgusted by the condition of homes with children than ones with animals. I also have known people who had neither children or pets who could take the Grossiest Home Award hands-down.