The goal is for every American citizen's vote for President to count for exactly as much as any other citizen's. That means, yes, the person who gets the most votes wins. How else can it work?
I'd encourage you to take a look at the National Popular Vote's site, especially http://www.nationalpopularvote.com/pages/answers.php . They've put quite a bit of thought into it.
Palestine has a parliamentary government. Their elections use a close relative to proportional representation.
Approval voting is for executive positions. President, Governor, Mayor. Not assemblies like councils and parliaments.
Approval voting prevents the "spoiler effect". It does not enable it. Nor is it applicable to parliamentary elections, such as the example you gave.
Lastly, your comment is just weird. Equating Hamas to a spoiler. Despite the electioneering by the PLA. The subsequent extra-legal attempt to invalidate the election results that brought Hamas to power.
I'm far from an expert on Palestinian politics, but if there's a lesson to be drawn from Hamas and The Gaza Strip, it's that Palestinian politics are messy.
If the question is "should they count the same" -- the answer is an unequivocal yes, and almost every person would agree. Voting reform requires that people put in power by bad systems willingly reform those systems which would see them removed from power. Corruption doesn't remove itself, so discussing this at all is moot, really.
Did you read what I wrote? That's not true of the National Popular Vote. Legislators in a plurality of states can alter the election for President (and so fundamentally shift the nature of national politics) without any cooperation from the President or any Federal official.
 Or rather, a number of states comprising a plurality of electors.
The system that puts legislators into place is a local majority vote, which is just as flawed as a national majority vote. It leads to a 2-party system which concentrates power and typically results in corruption.
Local majority vote --> Two-party system
National majority vote --> Two-party system
National majority vote <--> Two-party system
Two-party system --> National majority vote