There are definitely circumstances where it’s the best thing, and equally there are circumstances where MN go completely OTT.
I used to have an African grey parrot. For anyone who has never owned one, they are incredibly intelligent, demanding birds,needing almost constant stimulation. 5 years ago I went into heart failure following the flu, and at that point I was no longer able to give my parrot the attention he deserved. I was obviously reluctant to rehome him, but 6 months later I had a relapse and ended up in a position where I couldn’t even walk upstairs or from my lounge to my kitchen without becoming breathless,much less give the parrot the attention he needed.
I didn’t want to rehome him to someone who thought that having a parrot would be cool because they thought having a talking bird would be amusing etc, so I looked around for somewhere to take him.
He went to live with my parents for a while, because I spent time in and out of hospital, so living there he had more attention than he did here. And then we found him a home in a parrot sanctuary in the Lincolnshire wildlife park.
He now lives there with lots of other parrots, in an aviary, where he has the company of his own kind, the space to be a parrot, and isn’t going to randomly find himself back in the same position in the future.
Now here’s the thing. I had to rehome an animal in difficult circumstances and now I am much better. But it wouldn’t feel appropriate to go out and buy another parrot knowing that I already have a parrot out there who had to be rehomed somewhere else. It just wouldn’t. So while I wouldn’t say I should never have any other pet (although I don’t currently have plans for any,) I do think that having rehomed a bird it would be wrong to think now that things are better for me, I could go out and buy another one and so on. It’s not because I’m an irresponsible pet owner, if I’d kept him it would have been for my benefit not his, but because I’m a responsible pet owner I don’t get rid of my pets when the need arises and then move on to new ones when my own circumstances improve.
However I am reminded of that thread where an OP said that she didn’t love her dog as much as her baby, and posters were falling over themselves to tell her she should “rehome the dog and never have another animal.” 