[ty] Remove `ConnectionInitializer` (#19353)
## Summary
This PR removes the `ConnectionInitializer` and inlines the
`initialize_start` and `initialize_finish` calls.
The main benefit of this is that it will allow us to use
[`Connection::memory`](https://docs.rs/lsp-server/latest/lsp_server/struct.Connection.html#method.memory)
in the mock server. That method returns two `Connection` where one of
them will represent the client side connection and the other will be
sent to the `Server::new` call to be used by the server. This way the
mock client can send notifications and requests to mimic the editor.
## Test Plan
I tested out the initialization process and checked that the initialized
result contains the server capabilities and server info.
