Simple and pure mathematics is what keeps the coins safe, in order to make it harder for quantum computers we just need more complex math/equations.
Enigma in WW2, was the ultimate crypto/math problem, but the first computer invented managed to break the encryption, now it's the other way around, we need to invent an equation so the new computer generation can't break it.
Let inventing such thing to cryptography expert. Adobe (which is big company) tried building their own cryptography, but ended in huge failure[1].
Can anyone explain the final bit about transitioning to a new algorithm not being too difficult?
If ECDSA will be broken (and only that), then we can just create a new address type, and move all coins there.
There are millions active addresses. The process of moving coins from them would be very long and very expensive.
That's probably only true for company and individual with complex wallet setup. Most people just need to wait their wallet software/hardware to support new address format, then they could just send their coin. I would worry more about security risk when people creating new wallet and move their coin.
[1]
https://nakedsecurity.sophos.com/2013/11/04/anatomy-of-a-password-disaster-adobes-giant-sized-cryptographic-blunder/