Use public API for accessing aioesphomeapi client loop

[puddly]

esphome/aioesphomeapi#1598 has been merged so we can drop the internal _loop access.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 90.48%. Comparing base (765e856) to head (6a98122).
⚠️ Report is 2 commits behind head on dev.

Additional details and impacted files

@@           Coverage Diff           @@
##              dev      #69   +/-   ##
=======================================
  Coverage   90.48%   90.48%           
=======================================
  Files          19       19           
  Lines        3143     3143           
=======================================
  Hits         2844     2844           
  Misses        299      299           

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