People are sheep. Herd animals. Armies typically move in one direction without having to be instructed to. And so do leaderless crowds.
And this is especially true for people in politics. They need to move in groups, large groups preferably, because without the numbers represented by such large groups, they have no power.
So, the 'label' is actually REALLY IMPORTANT. Not just because some people fit into category X or Y, but because they IDENTIFY with the label, and let the 'ideology' coming with the label SHAPE their opinions. Why ELSE would you see such a strong correlation between people that, say, a) Want low taxes b) Are christians c) Are 'tough' on immigration d) Want the USA to have a stronger military? There is no inherent reason for someone who wants low taxes to want a stronger military (after all, that has to be paid for with taxes) or for a christian to want a stronger military ('turn the other cheek') or want low taxes ('give unto the emperor') or be against illegal immigrants (Jesus WAS an illegal immigrant himself, probably). So, why do a lot of these views form a related cluster with many Americans? Because of the label/party ideology.
This is especially true for the political animal, the career politician. Sure, you may have your Ralph Naders and your Ron Pauls, but the obvious way to get ahead in a political party is to tow the party-line, to 'fit the bill' perfectly, without many dissenting opinions.
And such career politicians are generally very keen to keep their fingers on the 'pulse' of their party, to follow what they feel is the consensus within it. USA parties may not have 'chief whips'
like the UK parties do, but they do have party organisations that try to make everyone on a ticket of their party fall in line. But even without those, people would be alligning themselves to 'fit' with the current party-line. And if that 'line' is uncertain... under internal discussion, they will become careful, go silent on it. This 'mechanism' is pretty much a given for any political party, anywhere.
So... as much as you may resent 'labels'... you cannot understand politics in modern democracies without considering them.