Yes, I would recommend no home equipment.
Injuries on home equipment are very common, at the gym when coaches supervise their kids they can see the subtle changes in technique creep in, when a gymnast is fatiguing or developing a bad habit and they stop the skills before the injury occurs. Overuse injuries are also very common, doing skills over and over at home without technique getting corrected puts a lot of strain on certain body parts and this leads to long term persistent injuries that are so hard to heal, often take years.
Home equipment also encourages gymnasts to try things they haven't learned or are only just learning. Some kids are great and never throw skills unless their coach says they are ready, but often those things change when friends are over.
Air mats are my biggest concern, they can result in catastrophic injuries. The bounce in unpredictable, how bouncy it is will change depending on the amount of air in it, and that will change depending on the weather as air expands or contracts. If they hit the wrong spot on the airmat they go flying and that can end very badly. It also really encourages kids to try things they haven't been taught or might not be ready for, and using the airmat too much promotes poor technique, as it will throw skills around for them that are not being done correctly. They will be able to land it, and think they have it, but the airmat is doing it and they are practicing bad habits.
Having said all that, regular ,mats are a good idea. Gymnasts will be upside doing skills everywhere anyway and without mats they will do it on less forgiving surfaces.