I've always hated how our mounts poof in and out of reality when we ride them. They should persist in the world while they're toggled. But instead of acting like any other pet, how about giving them a use when not in the saddle. Make them a combat pet when dismounted.
This would give every player the ability to have an active pet for every class. Imagine your Senche tearing into a Dremora, or your Guar taking a big chomp of that Bear. Even the Horse in Skyrim courageously fought for you.
The damage mounts do can be piddly compared to a Sorcs Familiar, but this would be a huge QoL update for the sake of immersion, as well as a neat combat addition.
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