My sister has outdoor cats and an indoor cat, but no indoor/outdoor cats. There are a few reasons.
1) When cats go outdoors, they bring in insects that can get into carpeting as infestations. Fleas and ticks are pretty nasty and hard to keep them out when a pet can go in and out. They also often bring back dead birds and mice and some people have issues about a mouse head on the floor.
2) Another issue is cat goes out and doesn't return. Whether killed or injured by a car, predator or dog, shot by a person, stolen, or picked up by a county animal control, you need to be prepared that the cat just disappears. If outdoor cats disappear, you expect it happens. Indoor/Outdoor is losing a pet.
3) Often, neighbors don't appreciate animals wandering around their property. Some purposely poison them.
4) Indoor/outdoor cats must be extra identified and fully vaccinated.
5) There could be local laws against it. Any pet out without a leash can be illegal.