chore(letsgo): probe provider availability using actual instantiation and model listing (#5932)

Replace HTTP endpoint probing with real provider instantiation and model
listing to confirm configurations work and models are available. This
ensures that detected providers can actually be used, not just
reachable.

Changes:
- Refactor _probe_endpoint() to _probe_provider_availability() with
provider registry lookup.
- Instantiate actual provider implementations using config factories
instead of making HTTP requests.
- Call provider.list_models() with asyncio timeout to confirm
availability.
- Add environment variable substitution for config templates
(${env.VAR_NAME:=default}).
- Inject required provider attributes (__provider_id__,
__provider_spec__, __provider_config__).
- Implement specific error categorization for authentication failures.
- Clean up provider resources via aclose() after model listing.
- Show pip packages needed when provider dependencies are missing

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Signed-off-by: Matthew Farrellee <matt@cs.wisc.edu>

Author: mattf