Sick pets can be annoying, but they're practically IMPOSSIBLE in rotational play

I was having the same issue until I decided to load my Sims inventories with the wellness treat for pets. Anytime I revisit a household with pets and there's a sick(s) pet(s) I give them the treat and voilà! No more trips to the vet.
But yeah, it is really annoying. Even giving the treat is annoying since they need to spend the simoleons on that and I also don't like seeing my households' pets sick and dirty all the time.