refactor: move Database to client crate behind testing feature

[tisonkun]

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What's changed and what's your intention?

This reverts #3820 and instead move Databas to client crate behind testing feature.

This supersedes #4059.

Checklist

• I have written the necessary rustdoc comments.
• I have added the necessary unit tests and integration tests.
• This PR requires documentation updates.

[codecov]

Codecov Report

Attention: Patch coverage is 61.16505% with 40 lines in your changes are missing coverage. Please review.

Project coverage is 85.01%. Comparing base (097f62f) to head (383e571).

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #4059      +/-   ##
==========================================
- Coverage   85.41%   85.01%   -0.40%     
==========================================
  Files         985      987       +2     
  Lines      170616   171031     +415     
==========================================
- Hits       145724   145403     -321     
- Misses      24892    25628     +736