Slobodan is correct about people switching parties to vote in primaries. In an open primary, you may vote for whichever party candidate you wish (but you can only vote for one person.) In other words, you may vote in the Republican primary or the Democratic primary, but not both. In states with open primaries, occasionally you'll have campaign situations in which, say, the Democratic candidate has a lock on the election, but the Republican race is close, between a moderate and a very conservative candidate -- in which case, Democratic voters may be encouraged to vote in the Republican primary, in hopes of electing the more moderate candidate, so no matter what happens in the final election, the conservative will be shut out. In a closed primary, you have to register in advance, often well in advance, to vote in a primary. Some states don't have primaries. The Trump campaign is trying to eliminate primaries in states where he controls the local party, but might lose a primary. (e.g. Arizona.)
About our electoral college. It has its disadvantages -- a candidate can win without a plurality of the vote, as Trump did. In some cases, neither Presidential candidate wins a *majority* of the vote, because third parties do soak up a few percentage points. Gore would have won the election in 2000, rather than Bush, if a relatively strong third party candidate (Ralph Nader) hadn't soaked up quite a few normally Democratic votes in Florida. One reason to continue the electoral party is to assure that all of the US in covered by the candidates. There are vast swaths of America that would never see a candidate if not for the electoral college --- why campaign anywhere between the Mississippi and the coastal ranges when the distances are large and the population is sparse? Better to focus on the coasts, where your buck buys much more head count. But that interior area, when included in the electoral college, can move elections, as it did with Trump.
Parliamentary systems have some advantages over our executive system, but they may also develop really crippling disadvantages, as is evident in the current situation in Israel, where very minor parties can demand, and get, pay-offs to their small minority positions. In Israel, it's been basically the extremely conservative religious parties that have kept Netanyahu in power, even though a large majority of Israelis, including most of Netanyahu's larger-party allies, strongly disagree with the demands of the religious parties. But, if you want the power, you have to pay them off. That mostly happens in narrowly balanced parliaments -- as Britain is finding out now, trying to deal with Northern Ireland on the Brexit issue.