A Republican form of government has its leaders dining in topless bondage nightclubs.
Oh, wait! My bad. That's a Republican Party form of government!
In all seriousness, the problem is that "republic" is a word that is very very broad. It doesn't mean what some people think it means. The Dictionary.com definition is this:
1. a state in which the supreme power rests in the body of citizens entitled to vote and is exercised by representatives chosen directly or indirectly by them.
2. any body of persons viewed as a commonwealth.
3. a state in which the head of government is not a monarch or other hereditary head of state.
4. any of the five periods of republican government in France.Compare First Republic, Second Republic, Third Republic, Fourth Republic, Fifth Republic.
5. a philosophical dialogue (4th century b.c.) by Plato dealing with the composition and structure of the ideal state.
So really, being a "republic form of government" has nothing to do with an electoral college. Getting rid of it wouldn't make us no longer a republican form of government in the slightest.