In the past, many people who would normally be considered as handicapped have served in militaries around the world. Most of the time, this happened in wartime where wounded people were kept in service because of their experience. There are some other exceptions that I know about.

During WWII, there was a legendary British fighter pilot named Douglas Bader. He had lost both legs in a crash in the early 1930s. When the war broke out, he fought his way back into a cockpit because the RAF needed experienced pilots. He rose in rank to Group Commander, became an ace, and ended the war as a POW following a mid-air collision. I just missed getting to meet him back in 1979 by an hour, something I regret to this day. He died a year or so later.

Following the war in Viet Nam, quite a few handicapped veterans were allowed to keep serving in the Army if they wanted. Normally, this was allowed if the soldier had a distinguished service record, typically requiring a Silver Star.

The movie "Men of Honor" is based on a true story about a Navy diver who lost a leg and was allowed to continue to serve.

There was a member of the Army Golden Knights parachute team who lost his legs in a mid-air collision years ago. He worked hard to pass the physical exam (after a tremendous amount of rehab) and became the first double amputee allowed to reenlist.

As for allowing people with preexisting medical conditions to enlist, I don't know how well that would work in practice. There were a lot of jobs that I did in the Air Force that could easily have been performed by someone in a wheelchair. However, most military facilities around the world just aren't set up to handle handicapped people, nor would a lot of places be able to handle people with special medical requirements.

The Navy has to spend millions to retrofit ships to handle female crewmembers. Imagine how much it'd cost to make the ships handicapped accessible. If you said that handicapped people wouldn't be eligible for shipboard service, then you're going to permanently give them a shore billet. That reduces the chances that people now serving at sea get to fill a shore billet, and that's really bad for morale.