The data is pretty clear that a truly random sample can indeed be representative of a larger population if the sample size is large enough. The larger the sample size, the more accurate the results (i.e. the smaller the margin of error). Any objective test for statistical significance shows that a sample size of 1,000 or 2,000 can be representative of 319 million with a reasonable margin of error. I would expect that virtually every statistician agrees on the power of random sampling. Aggregate polling has a great track record in predicting the results of elections, particularly when they're just before an election.
You can do a test yourself to gauge the power of random sampling. Let's say I am about to flip a coin 319 million times. Pretending we don't know the ratio of heads to tails in a coin flip, how many coin flips would you have to do in order to have a satisfactory random sample to figure out the likely result of my 319 million coin flips? The answer isn't even close to 319 million.
However, you do bring up a great point about how polls are very often treated as jerk-off material. A poll is only a snapshot in time (i.e. results may change wildly just as public opinion may change), and not all polls are created equal for the reasons I explained above. Therefore, it's easy for someone on one side of the political isle to engage in confirmation bias and only look at the polls that tell the story he or she wants to hear. If one cares more about what's true and less about what helps him or her sleep at night, it's important to look at careful aggregates and not focus too much on any single poll. Outliers will always exist in one direction or another for many reasons I've already outlined.
For example, an Ipsos poll came out recently showing Secretary Clinton ahead of Donald Trump by three points. The poll's methodology was good, and it had a sample size of a whopping 2,434 likely voters. In summary, the poll did just about everything right. However, to say this means Clinton is ahead by 3 would be to ignore all the other polls before it. In a good aggregate that gives more weight to polls that were done recently, polls with higher samples sizes, and polls with good methodology, Clinton is actually ahead by about 1.9, not 3. What would be significant is if polls consistently started showing Clinton ahead by 3 again, and the aggregate number would reflect that. With an aggregate number, we're also dealing with many more thousands of people in the sample size.
