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 94.28571% with 10 lines in your changes missing coverage. Please review.

Project coverage is 76.24%. Comparing base (4d8009d) to head (f84cf99).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #8940      +/-   ##
==========================================
- Coverage   76.28%   76.24%   -0.04%     
==========================================
  Files         463      464       +1     
  Lines       70474    70516      +42     
==========================================
+ Hits        53760    53767       +7     
- Misses      13368    13395      +27     
- Partials     3346     3354       +8     

Flag	Coverage Δ
unittests	76.24% <94.28%> (-0.04%)	⬇️

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

[JmPotato]

@rleungx @okJiang PTAL, thx!

[okJiang]

rest lgtm

[ti-chi-bot]

[APPROVALNOTIFIER] This PR is APPROVED

This pull-request has been approved by: okJiang, rleungx

The full list of commands accepted by this bot can be found here.

The pull request process is described here

Needs approval from an approver in each of these files:

• OWNERS [rleungx]

Approvers can indicate their approval by writing /approve in a comment
Approvers can cancel approval by writing /approve cancel in a comment

[ti-chi-bot]

[LGTM Timeline notifier]

Timeline:

• 2024-12-25 08:32:48.534419925 +0000 UTC m=+1636958.623222453: ☑️ agreed by rleungx.
• 2024-12-26 08:05:29.068391698 +0000 UTC m=+1721719.157194240: ☑️ agreed by okJiang.