I'll post another reply, since this kept being on my mind. I think the reason this is bothering quite a few because from a game design perspective, with hundreds of dishwater pets of one species, the same name for hundreds of pets on end, there is absolutely no sense of progress. There is no goalpost to get to, no discernable end to it. I believe this is why it is so tiring.

Maybe there is a solution to this outside avoiding someone altogether, perhaps limit how many pets can be nurtured of one user in a day (let's say as an example number like 500), and their pets will not appear anymore unless you find no more results - that way people who target nurture for pouches can just pick the user and nurture their whole hatchery, but for others who want to do the whole site it will be until they nurtured all else before they will get back to the mass-breeding users. That would mix things up and make it a bit more fun/progressive :D

How about a "skip to next hatchery" button? Or even arrows on the page that let you navigate through the nurturing grounds. A +1 pages arrow, +10 pages arrow, etc.