I think that the public would trust Crypto-currencies if they were not so volatile. No-one wants to see 50% of their investment suddenly wiped away by a few tweets - but Fiat doesn't drop like that just because of a few tweets or government annoncements.

The key to a more stable crypto-currency is for more shops (goods and services) to accept it. Then people wouldn't have to compare it to "fiat" equivalent and it might stabalise a lot more.

It will be a long transation period - maybe 5, 10, 50 years - but I think that the principle of blockchain will prevail.