From a player perspective, I'm pretty lenient on death as a feature. It is a fact of life, and could add realism to the game. If the game is not meant to be completely realistic I would love a way to keep my favorite pets alive, perhaps through an item, but it isn't a deal breaker for me personally. It could help keep a larger audience since players know they can keep the pets that are sentimental to them, and their projects or pets they spent a lot of currency on wont die. As long as it is done respectfully and doesn't have overly graphic text, just a simple "X has passed away," then it doesn't seem like it's be too harsh, particularly if you are looking for an older audience. A game for young children, I can definitely see it being a problem if death were to be added to a game, but a 13+ audience should be able to handle the idea.
From a very novice developer perspective, death or another way of cleaning out the data base is essential, especially when the focus of the game involves breeding lots of pets. Let's use wajas as an example. The youngest pets' IDs are around 8.6 million. What if this site didn't have a way to remove pets? They have to have a database entry for each pet and generate each image separately since they have no limits on markings and mutations, hex color codes, and opacity, and there are players who collect pets with as many markings and mutations on them as possible. That's a lot of data!
If you don't want death to be in the game due to sensitivities or any other reason, there are always other options to reduce the load. Flight Rising has exalting dragons to serve deities, Wajas has the Obsessed Waja Fan, and Novilar has releasing as Digital said. I have to say, I checked out the wiki for Novilar and the text displayed when you release a pet is absolutely beautiful. I love the idea of having releases being the pet's choice and is a wonderful alternative to death. The text also involves lore surrounding the site and it is my favorite example of giving players a feature to remove unwanted pets from the site without the pets dying.