Just like gold, in fact, during a recession and bank failures Bitcoin may just perform very well, because it has limited quantity, it is decentralized, and it is relatively decorrelating from traditional financial markets.
Like some other members I think that you answered your own question... so the final answer would be yes. Bitcoin will survive economic collapse, and I am sure that some other cryptocurrencies will survive as well. Some of them will even thrive, like Bitcoin, and maybe Ethereum, or some others I don't know, nobody knows... But I guess we all have some picks, but time will tell who was right.
Traditional financial markets are in problem... it's a fact. New digital age is here and it has something to offer...