Bitcoin is a social phenomenon as much as it is a technical one. Satoshi's whitepaper laid out a technical architecture for a network, but the implementation details of that network have slowly evolved over time, they are not absolutely set in stone.
Of course, some of the implementation details are perhaps more set in stone than others (eg, good luck trying to convince bitcoin users to change the issuance schedule), but ultimately it is only the collective agreement of the network participants that upholds the 'consensus rules' (ie, the implementation details).
So, if something catastrophic were to happen, and ECDSA were broken in a way which made all coins vulnerable to theft, then I would certainly expect that bitcoin users would work to change the implementation details, so that they could form a new social agreement around using the technical tool that is the bitcoin network.
Would this be messy if it happened unexpectedly? Most certainly. Would the value of the network drop? Probably. Would it mean that the bitcoin experiment would be over? I suppose that depends on what you define as 'bitcoin'. From a "no-hardforks" perspective, perhaps bitcoin would die (ignoring the possibility of a fancy soft-fork fix). But from a "we're here to participate in a novel consensus network" perspective, it certainly is possible to fork the network to change the implementation details in order to remove the compromised code. Whether or not you're happy to participate in this new consensus is a fair question, but I doubt every single bitcoin user would just give up and pack it in.
More generally, if public key cryptography is broken, I think the world is going to have bigger problems than the bitcoin network failing to remain secure. A huge amount of the (very valuable) internet is built upon the same cryptographic primitives that bitcoin uses. In spite of this, these algorithms have apparently remained secure. Ultimately, whether or not you participate in something like bitcoin is entirely up to you. I don't think anyone here will try to convince you it is a risk-free adventure to embark upon.