but how can you be certain about the validity of the chain without verifying all transactions?
By verifying a proof, and making sure, that faking a proof is more difficult, than overwriting the whole chain. For example: if a given proof is valid, and can be faked after making 2^128 SHA-256 hashes, then it is probably safe enough to be used in practice, because the total chainwork is a number around 2^96. And if it will reach something around 2^128, then SHA-256 may no longer be collision-resistant, and we will need to make a hardened version of it (for example like SHA-1 was hardened in the past), or introduce a new hash function, and validate new coins with that.
What ZK-proof do you have in mind?
It doesn't have to be ZK-proof specifically. Many models are possible, now it is more about encouraging people to think about different solutions, than pointless filters. And then, if enough people will think about it, then they will notice, that making such things is possible. Otherwise, some altcoins wouldn't exist, if it would be impossible. There are many chains, where you don't have to know everything, or where you don't know, who is sending what and where. Bitcoin can also use such features, it is only a matter of writing some implementation, and convincing the community, that it is better, than for example forking a network for no reason.
there's only one way to fully validate the current utxo set and that is by downloading the entire blockchain from the genesis block.
If that we're true you'd have a point, but the good news is that isn't.
It's possible using cryptography to construct proof for statements like "0xDEADBEEF is the hash of the tip of a blockchain starting at the genesis block where all rules pass, with total difficulty Y", where the proof is much smaller than the blockchain (in some cases only a few hundred bytes).
Such systems are already in production use for small programs today. Scaling them up to work over the whole bitcoin blockchain is a (considerable) engineering exercise, but I think it's inevitable-- well inevitable that the proof systems are developed to that extent. If Bitcoin will deploy them or not will depend on if anyone is still willing to work on it.
(And you should hope these tools are developed, because we've already seen what people do when validating the history becomes too expensive-- they skip it)
Also see for example sudoku proofs:
https://github.com/zcash-hackworks/pay-to-sudokuIn a similar way, as this sudoku proof, other things can be proven. You can have a bunch of arbitrary computations, and make a proof for that. For example: that the n-th Fibonacci number is equal to x. Or that a given number is prime, with 2^-256 probability of being composite, which is why elliptic curves can be generated at all, without checking 2^128 numbers, to make sure, that some 256-bit number is actually prime.
And if you read about basic proofs, and how they work, then you will notice, that validating the full chain, can be expressed as just a lot of simple computations. If you can construct a proof for all of that, and if you can combine proofs recursively, then you can prove anything, with arbitrary probability. And then, you will have a choice: to accept a proof, or to think, that someone has more computing power, than needed to overwrite the whole chain, and that some sneaky hacker found a way to fake proofs, and didn't abuse that power, to claim all 21 million coins instead, even though it would be N times easier task.
