Browser extension: Make sure BExt works post 4.0

Created by: philipp-spiess

This is currently blocked by whatever we do for #38317 (closed).

We need to change the browser extensions to not use Sourcegraph extensions but the new code navigations APIs that will be used for #38317 (closed).

Either use the new code intel APIs or load only the code intel extension for instances post 4.0

/cc @muratsu @jjinnii @ryankscott