Reuse target client in doc reindexer

[peternied]

Description

Small refactor for testability of document indexer actions, following from the snapshot test cases [1].

• [1] Basic snapshot tests #664 (comment)

Check List

• New functionality includes testing
  • All tests pass, including unit test, integration test and doctest
• New functionality has been documented
• Commits are signed per the DCO using --signoff

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[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 75.45%. Comparing base (c8be42b) to head (24a9826).
Report is 9 commits behind head on main.

Additional details and impacted files

@@             Coverage Diff              @@
##               main     #672      +/-   ##
============================================
- Coverage     75.51%   75.45%   -0.07%     
+ Complexity     1422     1419       -3     
============================================
  Files           150      150              
  Lines          6147     6147              
  Branches        561      561              
============================================
- Hits           4642     4638       -4     
- Misses         1135     1139       +4     
  Partials        370      370              

Flag	Coverage Δ
unittests	75.45% <ø> (-0.07%)	⬇️

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[chelma]

LGTM