Remove asserts and add documentation following #1054

[jlowin]

Fast follow to #1054 to remove asserts (replace with runtime errors) and document the reentrant strategy. Related to #1051

[Copilot]

Pull Request Overview

This PR refactors the client’s session management to remove ad-hoc asserts, add detailed documentation, and implement robust reentrant async with support with reference counting and a background runner task.

• Added comprehensive class- and method-level docstrings explaining the reentrant context logic and race-condition fixes.
• Streamlined __aenter__, __aexit__, _connect, _disconnect, and _session_runner by removing inline assertions and centralizing error handling.
• Introduced an explicit runtime error when the nesting counter is invalid and ensured events are created only when needed.

Comments suppressed due to low confidence (1)

src/fastmcp/client/client.py:329

• The new guard against a nonzero nesting counter when starting a session should have dedicated tests to verify both the valid reentrant scenarios and the error path when the counter is unexpectedly nonzero.

                if self._nesting_counter != 0: