I watched the entire thing. They were trying to go for a Mystery Science Theater 3000 thing, but it was extremely annoying and most of their points were bad.
With that said, we SHOULD keep the electoral college, but reform it.
There are actually two main points to the electoral college:
(1) Maintain a balance between big state America and small state America. The trade-off so to speak was to create a system where the larger the population the more electoral college votes/congressional districts, and the smaller the population, the higher the vote per capita. Lets take the extreme as an example. Wyoming has about 577k residents and 3 electoral college votes, representing the 1 congressional district and 2 U.S Senators. California has about 40 million resident and 55 electoral college votes, representing the 53 congressional districts and 2 U.S Senators. If you we do the quick math, the vote per capita in California is about 72k, and 192k in Wyoming. In other words, a vote in Wyoming is worth about 2.5 times a single vote in California. This was not by accident, but to create a sense of balance: Live in a small state, your vote gets count a little more; live in a big state, your state gets more representation.
(2) This is becoming more and more irrelevant, but it is an important wing of electoral college system. Our framers believed in a checks and balance system. The EC was setup so experts could review the people's choice and see if the assigned candidate is actually qualified to be the next President. With technology becoming more and more present and a more educated population, more states are leaning toward making it illegal to be a faithless voter and voting for somebody they were not assigned.
Where I believe EC goes wrong is when states decided to follow WV's lead and go for winner-take-all systems. This created a system where it doesn't matter if you win by 2 votes or 2 million votes, you're still awarded the same projected EC votes. A better system would be to divide out the number of electoral votes by proportionality. As in, you win 42% of the vote, you get 42% of the electoral college votes. If there's any leftover electoral college votes, then you give it to the winner of the states popular vote.