Use remote-aware skill locations in UI consumers

[moirahuang]

Description

Linked Issue

• The linked issue is labeled ready-to-spec or ready-to-implement.
• Where appropriate, screenshots or a short video of the implementation are included below (especially for user-visible or UI changes).

Testing

• I have manually tested my changes locally with ./script/run

Screenshots / Videos

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[moirahuang]

Warning

This pull request is not mergeable via GitHub because a downstack PR is open. Once all requirements are satisfied, merge this PR as a stack on Graphite.
Learn more

• Use remote-aware skill locations in UI consumers #11581 👈 (View in Graphite)
• Support remote project skills #11459 : 1 other dependent PR (#11460 )
• Move local and remote project skills to use RepoMetadataModel #11559
• master

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