My cats were indoor cats- the fact is a cat allowed to go out has less than half the life span of a cat left indoors.
The cat wants to go out because there are things to do... You wont believe this- but I set the house up. I put streamers like used on pom poms on the vent covers so when the heat or ac went off they would blow- .. I had tons of cat toys of feathers and played with the cats when I was watching tv- or even drinking coffee... As long as they can use that natural hunting ability- they are happy. I also had ping pong balls as I had hard wood floors- they loved those too...