Give the Rust portion of Synapse access to the database

This issue has been migrated from [#16501](https://github.com/matrix-org/synapse/issues/16501).

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Eventually it seems like we will want to get the Rust parts of Synapse some access to the database so we can offload some of the expensive processing (including state resolution) to Rust, sidestepping the GIL and likely improving performance generally.

This is not exactly trivial.

## Option 1: expose existing connections to Rust (i.e. Rust calls Python)

Good:
* reuses our existing, well-understood database machinery

Bad:
* calling Python whenever Rust wants to access the database means we will have to lock the GIL for database accesses (and there is probably other interop overhead too)

## Option 2: move database machinery into Rust and have Python call Rust

Good:
* Rust doesn't have to lock the GIL for database access

Bad:
* probably means implementing a DB-API2 wrapper around the Rust client libraries (not convinced this is very hard though, but beware the unknown)
* sacrifices our well-understood machinery; we won't necessarily be immediately confident in the transition
* likely interop overhead from Python calling Rust (which will be the majority of the time unless we dramatically move all our DB transactions to Rust) — that said, we already pay this cost moving C-struct Postgres results across to Python, it's just done by psycopg2. It would be quite interesting to do a benchmark of a naïve Rust DBAPI2 wrapper for Postgres/SQLx/something against Psycopg2 and see if this would hurt our perf or not.

## Option 3: maintain connections in both languages

Good:
* No interop overhead

Bad:
* We would now have two versions of each database connector in play
* It's probably impossible to use `:memory:` SQLite databases (e.g. in testing) this way. Though we could put the file on a tmpfs as a likely suitable workaround.
* Someone has to decide the connection limits for both connection pools and it's not going to be obvious to us or to end-users what that split should be.

## Any others?