I think it's important to point out that not only did Twilio choose a great problem to address, it was (critically) also one that a large audience already understands is very difficult. There are plenty of equivalently "valuable" problems to solve, which aren't as easy to communicate effectively, because most people lack enough domain-specific knowledge to understand the difficulty of the problem.

This, I think, is the core truth behind the oft-repeated observation that you should "pay attention to the startups with a few customers that absolutely LOVE their product". Not all valuable problems are ones that a general audience is likely to comprehend, but when you narrow it down just to the audience that does, how do they feel about what you're doing?

Put differently: the Twilio demo is so effective not just because they have a very good problem, but because it's an exceptionally common one as well. They have an impressive solution to a tough problem, sure, but they laos did a very good job aligning "the people who have this problem" with "the people listening to this demo", and that's what makes it great.