chore: Switch legacy redirect handling to new pattern

[stevehipwell]

This PR switches the legacy redirect pattern that modifies the global client to use the new pattern introduced in #3411.

Related to #3915

[codecov]

Codecov Report

❌ Patch coverage is 87.50000% with 2 lines in your changes missing coverage. Please review.
✅ Project coverage is 93.85%. Comparing base (cd716ed) to head (9016f4e).
⚠️ Report is 5 commits behind head on master.

Files with missing lines	Patch %	Lines
github/migrations.go	60.00%	1 Missing and 1 partial ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #4161      +/-   ##
==========================================
+ Coverage   93.83%   93.85%   +0.01%     
==========================================
  Files         209      209              
  Lines       19685    19667      -18     
==========================================
- Hits        18472    18458      -14     
+ Misses       1015     1013       -2     
+ Partials      198      196       -2     

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[gmlewis]

Thank you, @stevehipwell!
Do we have any integration tests associated with these endpoints so that they can be validated that they functionally still work? If not, do you think this would be a good use of integration tests?
Otherwise, LGTM.
Awaiting second LGTM+Approval from any other contributor to this repo before merging.

cc: @alexandear - @zyfy29 - @Not-Dhananjay-Mishra - @munlicode

[stevehipwell]

@gmlewis TestRepositories_DownloadReleaseAsset covers DownloadReleaseAsset. I don't think migration would be something to include in the integration tests due to permissions.

[Not-Dhananjay-Mishra]

LGTM

[gmlewis]

Thank you, @stevehipwell and @Not-Dhananjay-Mishra.
LGTM.
Merging.