I totally understand your feelings. After my little dog Harry died in 2017, a part of me died with him. It was stressful and expensive going through his treatment for cancer, and then watching him die was so hard. We both decided that we will not go through that again. We miss having a little pet in the house but at this age we will just play with the neighbor's dogs.
I crossed paths with one of the folks that I was in Rebuilding Together with. Dick is in his early 80s. I was at his place when the neighbor's dog can running over, staring into the screened-in porch. Dick got out the treats he keeps for the critter, we petted it for a while, then it went home. We added "pets" to the things that it's better to borrow from friends than to own (the other things being pools and boats.)
I think if I were alone I might feel differently; a pet does help keep one from feeling lonely. Since my sister's husband died she has managed pretty well since she still has her little dog companion to care for.
We had a Fox Terrier who did that all the time. I imagine it's a hunting thing...it masks their own scent. Either that, or it's just to gross us out. In our household it made him one of the guys because my sisters were disgusted by it. That made us think it to be an endearing quality