Bitcoin relies on a bunch of crypto for it to work, far moreso than traditional currencies, commodities, or shares. It also relies on ubiquitous computing and internetworking, and the styles of thought that accompany those things.

Yes, it's underdefined and underproven from a professional or academic crypto perspective. But worries about such perfection slowed or discouraged others from building something similar or better before now.

In that way, Bitcoin's a bit like Linux or the Web: an ad-hoc success where more professional approaches had stalled, in development or adoption. It likely has some success-bringing aspects that even its designers and current maintainer don't fully understand, but lucked-into. Sometimes these things are only fully appreciated in retropect.

Also like Linux or the Web, a lot of formalization and hardening can be bolted on later, now that the essential utility is proven and the potential rewards for fixes are clear. Even if giant parts of the existing system collapse -- the SHA256-based mining, the P2P transaction broadcasts, the bloated global blockchain -- if the ledger of keys & balances survives, multiple teams will then try to reboot a next-gen system that allows those balances to keep trading.

(I conjecture that crypto control of 1 BTC today might wind up, a decade or two hence, spawning usable balances in a number of official and unofficial offshoot projects... vaguely like how 100 shares in the original AT&T eventually wound up becoming shares in both AT&T and the 7 'baby bells'. We will see 'baby Bitcoins' in the future, which attempt to accelerate their adoption by endowing everyone wiht a Bitcoin balance a balance in the new system as well.)