API tests are non-functional

[bruceg]

The API tests in Vector are gated behind the feature flag vector-api-tests. However, actually building the tests by enabling that flag (cargo test --lib --features api,vector-api-tests) results in 45 build errors. Obviously these should be fixed.

Note: This was discovered while resolving issues with the Rust 1.80 version bump (#20949). While resolving this issue, we should also merge the tests that were moved to lib/vector-tap/src/lib.rs back into src/api/tests.rs since the former rely on features that are only present in src.