Why does a currency need to prove it self? Did PayPal ask the world if they can solve the global warming problems or if they could cure hunger? People look at the use case for a currency and they apply it to improve on the previous option that they had and that is where Bitcoin comes in.
Bitcoin is not centralized, so political agendas cannot interfere with the payment process and it is pseud anonymous to protect the owners financial data. Bitcoin is also border less, so you can send bitcoins to anyone, anywhere in the world. People will look at that are use that in their own circumstances and that is how it solves problems and challenges.
Totally different beasts. Paypal is a centralized payment processor for fiats alike. Bitcoin's use case is extremely niche for the common average joe, that is why I think it will never be mainstream, or at least not as a regular means of payment. As long as you require technical expertise to secure Bitcoins, and this is a fundamental premise, then it cannot be mainstream by definition. For other reasons as well such as any business that's legal needs to go through KYC and regulations so people would rather get paid in fiat.
If a country is developed then its citizens aren't going to flock to BTC.
The main point is that BTC does not need to be mainstream to be worth millions of $, this is something that a lot of people seem to miss.