IMO, anyone that is in their 60's that has a mortgage is doing something terribly wrong.
I don't know that I'd go that far, but I certainly wouldn't be buying a house in my 50's, that's for damned sure.
As it stands, my mortgage isn't over until I turn 70. However, I know damned well that when I have the debt from the divorce paid off, I'm doubling up on house payments so that I can 1--max out 401(k) within 2 years, 2--KEEP it maxxed out, including the over-50 "catch-up" provisions, 3--be debt free by age 60, 4--NOT have to tap into a reverse mortgage, 5--get to the point where daily bills are LESS than SS, meaning home equity, cash balance, and 401(k)/Roth funds are gravy.
Nobody have ever left me anything, and I want to ensure that should anything ever happen to me between now and 60, Scoobie and the kids aren't going to have to worry about how to get by.