So I am thinking this:
If I can just store the complete set of all current UTXOs, plus the most recent 2016 blocks, then it is to all intents and purposes the same as running a 350GB full node.
How would you plan to get all current UTXOs without downloading the blockchain from the beginning?
It's simple: if I bought BTC in 2010, that UTXO was mined and recorded in the blockchain back then. If you only download the latest blocks, you will never know about it and think that that address is empty. This is why you need to download everything to build the UTXO cache yourself and verify everything.
If you don't mind verifying everything, just run an SPV node. It's as simple as that

If you don't have the storage for 350GB, I think pruned node is a good idea. It gives you many benefits over SPV, while only requiring a few GB of storage. Sure, the initial block download won't be faster than when doing a regular full node, but if you don't have bandwidth issues, it's really a no-brainer.
I'm not sure, what's the state on this, but it seems you can just run Bitcoin Core in SPV mode actually. Thus having access to all of the commands etc.:
https://github.com/bitcoin/bitcoin/pull/9076