Nope!
A succinct block chain does not in fact need to maintain a UTXO set. Coda for example is not UTXO-based at all; it uses versioned accounts. There is a Merkle-tree accumulator over account states. If you own an account then you need to store and update its witness in that accumulator; this is essentially the same as updating note witnesses in Zerocash/Sprout/Sapling. You don’t need or have access to other people’s account witnesses. If the bound on the number of account updates is N, then the size of state you need to maintain per account is logarithmic in N, so effectively constant [*]. The information needed to update your account witnesses can be included in each block, and is also of size logarithmic in N (it’s just the frontier of the Merkle tree). No-one needs any account witnesses in order to verify the block chain.
TLDR; it’s magic, and thoroughly solves the scalability problem, both asymptotically and practically.
The only significant caveat is the zk-SNARK trusted setup issue (since you need fully succinct zk-SNARKs for recursive composition), but look for further improvements to the trust requirements there, soon.
[*] If I’m being picky, logarithmic in the number of supported state updates is not constant, and so the claim in the introduction of the Coda whitepaper should be revised. But N = 264, for instance, supports a practically unlimited number of updates.