First off, I agree with the "Idea", bough mounts shouldn't die, but should at LEAST have a long wait time till re-use.
Glad to hear you're remaking it completely. As of the lastest update we can catch small animals/foxes but thy become an inventory item (I know this is just a placeholder) with skinning allowed, hopefully with the next update (or in a couple updates at least) we can take out a pet maybe even have a separate (NOT inventory) "Pet's window that also holds the "pet/item""
I would really like to see these features with the new taming system since its being reworked entirely.
A new UI for pets (or at least chat / commands for a placeholder until UI is made) with Passive / Aggressive / Kill/Attack (then pick target) / Guard mode (protects you) / Guard player (follows and protects another player OR tamed pet, until they go to far away then the pet returns in Passive mode (with a notification saying "Your pet has returned from (player/pet) because the distance was too far.).
I know the UI is a while to make so for now it'd be easy to do chat commands, /passive, /petkill (then a cursor comes up and you click on what to "attack", ect...
To make pets not static and boring have a stat system for pets (some are stronger but less defence) ect.. RND (within certain limits for pets) this would GREATLY increase the pet market, and the need to find the "perfect" pet.
Pet spawns, some pets like the cold (once you update your map with cold/desert (or "hot") mountains ect... yo could make certain pets only spawn there so it's an adventure to prepare for and not just "oh I came across (animal) time to tame,
You should have sub-species of pets, e.g. "Wolf", "Black Wolf" (likes caves harder to find ect..)
Keep up the good work!