Improve scheduling of write tasks on a locked socket

[philippjfr]

A continuation of #6028 that schedules tasks that write messages to the socket connection using asyncio rather than curdoc.add_next_tick_callback ensuring that they can be dispatched as soon as possible, while still avoiding the:

RuntimeError: _pending_writes should be non-None when we have a document lock, and we should have the lock when the document changes

error.

Fixes #5999

[codecov]

Codecov Report

Attention: 21 lines in your changes are missing coverage. Please review.

Comparison is base (aa61fe6) 84.68% compared to head (5010dff) 84.69%.
Report is 1 commits behind head on main.

Files	Patch %	Lines
panel/io/document.py	47.50%	21 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #6052      +/-   ##
==========================================
+ Coverage   84.68%   84.69%   +0.01%     
==========================================
  Files         295      295              
  Lines       43918    43929      +11     
==========================================
+ Hits        37193    37207      +14     
+ Misses       6725     6722       -3     

Flag	Coverage Δ
ui-tests	40.69% <47.50%> (-0.01%)	⬇️
unitexamples-tests	72.80% <17.50%> (+<0.01%)	⬆️

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