Cat keeps using doggy door

So my cat 6months has started to get to the backyard using the doggy door. Our backyard is small/medium and we are on top of a hill so there’s no real danger of him running to a busy road. He seems to just like the outdoors and when he is outside he just likes to sniff the plants and roll around.

I have been trying to leash train him and only take him out on a harness but it doesn’t seem like he needs it really cause he doesn’t try to escape or anything. There are lizards all over the backyard that he likes to chase.

Is it ok to just allow my cat to use the doggy door to go in and out as he pleases or should I really not allow my cat outside? If I’m gone all day working I’m afraid to leave the doggy door open cause he will go outside but then I come home to my dog peeing on the floor.

Yes he is microchipped but I had to take the “breakaway” collar off of him because it got stuck in his mouth and didn’t break away and caused him to bleed. Should I try another collar and put an AirTag on it?