client: introduce the connection ctx manager

[JmPotato]

What problem does this PR solve?

Issue Number: ref #8690.

What is changed and how does it work?

Previously, we used a `sync.Map` as a medium to propagate connection ctx updates between
the dispatcher and TSO client, which introduced a lot of redundant parameter passing and
made the logic less intuitive. This PR implements the same functionality using a common
connection ctx manager to simplify and reuse related code.

Check List

Tests

• Unit test
• Integration test

Release note

None.

[codecov]

Codecov Report

Attention: Patch coverage is 89.71429% with 18 lines in your changes missing coverage. Please review.

Project coverage is 76.27%. Comparing base (95bfbe6) to head (192f36e).

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #8940      +/-   ##
==========================================
+ Coverage   76.25%   76.27%   +0.01%     
==========================================
  Files         463      464       +1     
  Lines       70458    70500      +42     
==========================================
+ Hits        53731    53773      +42     
- Misses      13369    13375       +6     
+ Partials     3358     3352       -6     

Flag	Coverage Δ
unittests	76.27% <89.71%> (+0.01%)	⬆️

Flags with carried forward coverage won't be shown. Click here to find out more.

[JmPotato]

@rleungx @okJiang PTAL, thx!

[ti-chi-bot]

[APPROVALNOTIFIER] This PR is APPROVED

This pull-request has been approved by: rleungx

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[ti-chi-bot]

[LGTM Timeline notifier]

Timeline:

• 2024-12-25 08:32:48.534419925 +0000 UTC m=+1636958.623222453: ☑️ agreed by rleungx.