refactor: Update Client.run to have a better async I/O usage

[DA-344]

Summary

This PR refactors the Client.run logic to fix problems involving asyncio due to how the library used the loop:

Needs testing

Exception

  File "/home/container/main.py", line 23, in <module>
    bot.run(BOT_TOKEN)

  File "/home/container/.local/lib/python3.11/site-packages/discord/client.py", line 772, in run
    loop.run_forever()

  File "/usr/local/lib/python3.11/asyncio/base_events.py", line 608, in run_forever
    self._run_once()

  File "/usr/local/lib/python3.11/asyncio/base_events.py", line 1936, in _run_once
    handle._run()

  File "/usr/local/lib/python3.11/asyncio/events.py", line 84, in _run
    self._context.run(self._callback, *self._args)

  File "/usr/local/lib/python3.11/asyncio/selector_events.py", line 956, in _read_ready
    self._read_ready_cb()

  File "/usr/local/lib/python3.11/asyncio/selector_events.py", line 988, in _read_ready__get_buffer
    self._protocol.buffer_updated(nbytes)

  File "/usr/local/lib/python3.11/asyncio/sslproto.py", line 439, in buffer_updated
    self._do_handshake()

  File "/usr/local/lib/python3.11/asyncio/sslproto.py", line 560, in _do_handshake
    self._sslobj.do_handshake()

  File "/usr/local/lib/python3.11/ssl.py", line 979, in do_handshake
    self._sslobj.do_handshake()

Information

• This PR fixes an issue.
• This PR adds something new (e.g. new method or parameters).
• This PR is a breaking change (e.g. methods or parameters removed/renamed).
• This PR is not a code change (e.g. documentation, README, typehinting,
  examples, ...).

Checklist

• I have searched the open pull requests for duplicates.
• If code changes were made then they have been tested.
  • I have updated the documentation to reflect the changes.
• If type: ignore comments were used, a comment is also left explaining why.
• I have updated the changelog to include these changes.

[DA-344]

Applied all the changes Dorukyum requested.

Before merging, I would like some feedback on this discussion message