Can my panel support a 60A circuit for EV charging?
You're putting a 60A breaker in a 100A panel. That's fine.
The total of the breakers is more than the panel, because we assume every circuit will not be drawing full power at the same time. That's fine.
But if you use everything at once it may trip your 100A service. Unlikely, but possible.
When do you plan to upgrade to 200A?
Obviously you need to run 60A wire to your charger, and keeping that run as short as possible will save $$ in wire and voltage drop. Comment above about thicker wire is absolutely correct.
UNLESS code in your area requires a sub panel (I would have to look up the code) you don't need it for the electrons.
Free access to NEC at NFPA
You will NOT, of course, replace the 20/20 breaker with a 60/60 and connect to the old 12ga wire. But you knew that. That answer missed the point.