I began digging on the Archive nodes, and some people from the community did speak up and clear up some of my confusion. You can follow the conversation here (you'll need to be a Mina Protocol Discord member).
The tl;dr is that anyone can run an Archive node, meaning anyone can access the archives of the entire ledger.
These Archive nodes store block data and transactions, but not private data. Snapps will use proofs (zk-SNARKs) to prove that the data exists, even when users aim to keep the actual data private. That's where the "proving you know the solution to a puzzle without revealing the solution" analogy comes in.
The best example of how that works are the Teller Finance Snapps, which you can read about here.
Circling back to the Archive nodes, the only incentive to run such a node is to power a service that needs that data to function. Mina does not currently offer any rewards for running an archive node because the nodes are not necessary for the protocol to function. That seems like circular reasoning, but let's keep going.
Archive nodes seem pretty important, considering you need them to recover the blockchain if something goes wrong. Luckily, many participants are running Archive nodes, and not all of them are on Google Cloud. You can run these nodes anywhere you want, be it on Google Cloud, your own VPS, or in your basement.
A Genesis Founding Member put it this way:
Anyone can run an archive node, and many are. You also don't need trust, as we use proofs to show the current state is valid. Also, with the [Mina-replayer tool](https://garethtdavies.medium.com/running-a-mina-archive-node-f6299219aab1#:~:text=mina-replayer — replays transactions from the archive node.), you can replay the archive data (any) to produce a ledger from genesis corresponding to a known protocol state.
Another good samaritan delivered a lengthy response to some of my concerns here (again, you need to be a Discord server member). Below is a summary.