Is it possible for a country to have a popular political party that is firmly based on Objectivist principles?
If it is possible, then what would prevent people who do not fully accept Objectivism from joining the party and using the party as a vehicle for achieving political power?
Only if Objectivists principles attain popular support.
What prevents someone that isn’t “fully Repulican” from joining the Republican party and riding their platform to power? Political parties are largely a convenience, they are not and cannot be taken to be a defining characteristic. However, if someone lies to get votes and then doesn’t act to promote legeslation that falls under his platform, he will rapidly lose support and not be returned to office.
The American system can easily recover from a bad Presidency: it has before. It already has protections in place to prevent the excesses of corrupt politicians.