I'm far too young (60) to have any personal experience, of course.... But my dad, who died last year at 83, rode pretty regularly until he was 79. At that point he developed prostate cancer, which kept him off the bike during the initial treatment. He intended to start again, but then went through a long middle ear infection (not related to the cancer or treatment) that screwed up his balance.
The point, though, is that until his terminal illness, he was able to ride comfortably, so the things you (and I) are worried about may not happen to us.
If you do look for a new bike, you might consider something like a Rivendell Rambouillet or Atlantis (
www.rivbike.com). I have one of each (got the Atlantis new, then scored a great deal on a used Rambo). They both have room for big tires (at least 700x38), which makes them pretty stable, yet you can swap for skinnier ones and go as fast as you want. I weigh 240, and I use 700x35s on the Atlantis, usually 28s on the Rambouillet. Works great.