I'm also confused. There are a number of creative ways to prevent a cat from escaping. Many, many indoor cats will try to escape--that isn't unusual at all.
I also expected a serious behavior problem, allergy, etc., but this problem is very fixable! We'd love to help you brainstorm if you are still interested in keeping your cat.
|This cat will NOT stop running out the door, and where we live he is going to get killed.
That would indeed be upsetting but this problem can be solved, which is why people are confused.
Personally I would just secure the cat in another part of the house when I left. Second choice would be to use baby gates to create a "buffer zone" two feet in front of the door (so I had enough room to step inside, close the front door behind me, and open the baby gate in front of me). If you want to change the behavior, you could install a simple screen door hinged on the inside so you can open the front door, look inside to see if the kitty is waiting, and keep a spray bottle of water by the door to give him a blast through the screen until he backs off. Most likely that last one would radically diminish his interest in escaping! These were just idea's off the top of my head, and I would try them all, and many more, before thinking about rehoming the cat. The op didn't say what she tried already, so that is why people are confused and upset.