The more I think about it, the less concerned I am with whether or not he can physically do the action (stunt men and CGI can help a lot), and the more I'm more concerned about his personality as an older guy.
In KOTCS, we already saw a drastically different Indy. Instead of being the guy chopping a rope bridge in half, he was the guy yelling, "Somebody could get hurt!" and "This is intolerable!" Instead of the cool ladies' man we knew from the earlier films, he was an old man who clumsily stumbled over things. And, frankly, he just seemed like he wasn't all that into the adventure; he was merely a passenger, along for the ride.
Add another 10 years to that, and is it a character we really want to watch? Indy was popular because we wanted to BE him. As kids, we'd climb around on things, pretending we were Indy. Will kids today want to pretend that they're a 75-80 year old man who yells, "This is intolerable!"? I doubt it.
That's why I really think recasting is the way to go. Make Indy back into the younger "devil-may-care" guy we all loved back in the 80s, and set the movies back in the 1920s-1940s, not the 1970s.