"Greetings. You have been selected for the office of Senator..."
I have speculated on a "draft" type of method of picking leaders for some time. As you say, anybody who really, really wants to be president is probably not someone who should have the job.
I've fiddled with modifications on the basic idea. Maybe draft several people and have a run-off election to pick one. Something would have to be done about non-elected staff-type people so we don't end up with a permanent elite of "behind the throne" types.
Would it work better than what we have now? Probably wouldn't work any worse.