Add test --doc to xtask test and the CI

[snacks02]

Check rust-lang/cargo#6669 for context as it seems to be the issue here.

Nextest doesn't have the --doc flag, so the only option seems to be to use cargo test --doc.

Before:

---- libsql/src/lib.rs - (line 29) stdout ----
error[E0432]: unresolved import `libsql::Frames`
 --> libsql/src/lib.rs:32:24
  |
6 | use libsql::{Database, Frames};
  |                        ^^^^^^ no `Frames` in the root
  |
  = help: consider importing one of these items instead:
          libsql::replication::Frames
          libsql_replication::rpc::replication::Frames
          pprof::Frames

error: aborting due to previous error

[snacks02]

I see that this has been fixed in main as well, so I've updated this PR to only include the changes to CI and xtask.

[MarinPostma]

It seems your new CI step has uncovered a failing example ;)

[snacks02]

I wanted to update the PR, but was unable to because of https://github.com/tursodatabase/libsql/actions/runs/7621698836/job/20758481022.