A significant obstacle (probably not the only one) to making a Zcash-like block chain succinct, is Bitcoin scripting. It would be exceptionally difficult, and very inefficient, to verify Bitcoin-style scripts in a SNARK/NIZK circuit.
Pay-to-verification-key (P2VK) is a possible privacy-preserving alternative to scripting where an address specifies a verification key for a zk-SNARK circuit to be satisfied in order to spend from that address. It doesn’t have the above problem because the SNARK verification is a well-defined operation of a fixed size in the succinct-verification circuit.
So, the first thing I’d like to do to unblock the possibility of making the Zcash block chain succinct would be to remove transparent addresses. There are a few things that block that in turn; besides performance which is largely addressed by Sapling, you need to replace functionality that is currently only supported for transparent addresses such as multisig and scripting, and change parts of the consensus protocol that depend on them such as coinbase transactions and fees.