Allow me to propose an everyone-wins compromise; one that builds on your age 65 standard.
Remove all age restrictions on holding an office; Ten-year-old's could run for president along with 100-year-old's.
Then for voting, allow only those 65 and older to vote. This would allow for both the increasing wisdom and for the ability to spot real talent that comes with age. At the same time, it would eliminate the rank idiocy that comes with immaturity.
This way everyone would be allowed to participate, and the nation would be better run in the long run.