We had a situation a few years ago where my husband’s company was considering transferring him to Germany for a few years (military consulting). We didn’t have pets at the time, and the thing fell through for other reasons, but I have no idea what we would have done if we did have a dog. We probably would have been in base housing, or a very small town. I think Germany has some pretty strict BSL as well.

Back in the states, we actually bought a house because we could not find rentals that would accept dogs bigger than a corgi. We’re lucky we had the resources available to make that choice – most people certainly don’t. Even so, finding a house takes time – what would we have done with a dog in the intervening 9 months?

(but yeah, the woman in the post sounds a little off her rocker)

Other scenarios… Flying to your new home, in a season when the pet cannot fly; I don’t know about you, but I’m not sure I could afford to board a pet for months and then fly them out. In the same vein, moving internationally, especially on short notice; perhaps there is not enough time to meet quarantine requirements. Moving to an expensive area; perhaps they simply do NOT HAVE the money not only for the pet deposit, or the pet rent, or simply the more expensive rent at the only places that accept pets (I know in my area all the lower-income housing does not allow pets, so I’m lucky I make enough to afford my more expensive apartment, plus monthly pet rent, plus the hefty deposit I had to fork over when we moved in).
It’s nice to sit on a pedestal and believe there is no legitimate reason to give up a pet, but the reality is that sometimes you really truly cannot afford something, or cannot make it work logistically, and while in an ideal world that would mean you don’t take the new job, don’t move… in the real world things don’t always work like that. I agree that most people are really giving them up out of laziness/not wanting to pay extra/whatever instead of legitimate reasons… but legitimate reasons do exist. And as a note, there are families I WISH would give up their pet because when you can barely afford to feed yourself and your family, the dog gets short shrift and would probably be better off being rehomed.