[Bug]: Failed async startup leaves Connection.init task exception unretrieved

[karmas056]

Version

1.61.0 (latest release)

Steps to reproduce

Minimal reproduction: https://github.com/karmas056/playwright-init-task-repro

1. Clone the repository:

   git clone https://github.com/karmas056/playwright-init-task-repro.git
   cd playwright-init-task-repro

2. Build and run it:

   docker build -t playwright-init-task-repro .
   docker run --rm playwright-init-task-repro

The reproduction injects a minimal Transport that deterministically closes
while Connection.init is pending. It does not launch a browser or a real
Playwright Node driver, so it isolates the Python startup cleanup path from
browser and operating-system behavior.

The script exits successfully only after the event loop observes an unconsumed
Connection.run.<locals>.init task.

Expected behavior

The transport/startup exception should be raised to the caller, and every
internal task created by the failed startup attempt should be awaited,
cancelled, or have its exception consumed.

Garbage collection should not produce a secondary
Task exception was never retrieved warning.

Actual behavior

The original startup exception is raised to the caller, but after gc.collect()
the event loop reports an unconsumed internal init task:

startup_error=Exception: Connection closed while reading from the driver (forced repro)
result=REPRODUCED
task=Connection.run.<locals>.init
unretrieved_exception=Exception: Connection.init: Connection closed while reading from the driver (forced repro)
Task exception was never retrieved
future: <Task finished name='Task-3' coro=<Connection.run.<locals>.init() ...>

The relevant traceback ends with:

File "playwright/_impl/_connection.py", line 310, in init
    self.playwright_future.set_result(await self._root_object.initialize())
...
Exception: Connection.init: Connection closed while reading from the driver (forced repro)

Additional context

This was initially observed after a real, intermittent driver disconnect in a
long-running asyncio scheduler. A startup attempt logged
Connection closed while reading from the driver, retried successfully, and a
later garbage-collection cycle reported the abandoned init-task exception.

The deterministic reproduction points to the following startup path:

• PlaywrightContextManager.__aenter__ creates Connection.run() with
  loop.create_task(...) and waits for either on_error_future or
  playwright_future:
  https://github.com/microsoft/playwright-python/blob/v1.61.0/playwright/async_api/_context_manager.py#L29-L48
• Connection.run() stores a second task in _init_task:
  https://github.com/microsoft/playwright-python/blob/v1.61.0/playwright/_impl/_connection.py#L305-L314
• If the transport error wins, __aenter__ raises, so __aexit__ is not
  entered.
• Connection.cleanup() only cancels _init_task while it is still pending;
  it does not retrieve an exception from an already completed _init_task:
  https://github.com/microsoft/playwright-python/blob/v1.61.0/playwright/_impl/_connection.py#L327-L330

I searched the existing issues. #2091 has a related unhandled
Connection.run() startup task, and #650 covers transport errors during
context-manager initialization, but I did not find an issue covering this
completed _init_task cleanup path.

Environment

- Operating System: Debian GNU/Linux 13 (trixie), Docker
- CPU: x86_64
- Browser: N/A (the failure and reproduction occur before browser launch)
- Python Version: 3.12.13
- Playwright Version: 1.61.0