Migrate AiGateway::Client to Unit Primitives

What does this MR do and why?

Migrate AiGateway::Client and related classes to use Unit Primitives instead of AvailableServices. We are removing the latter as part of Phase 5: Remove AvailableServices interface (&15639 - closed)

I tried to manually check implementations to the best of my ability but I am unsure as to which use cases I may have missed.

Refactor, not user facing.

References

Related to Replace remaining AvailableServices.find_by_nam... (#564979 - closed)

Screenshots or screen recordings

n/a

How to set up and validate locally

This touches a number of Chat-based integrations. I verified to Chat to work as intended in general, but I am unclear on how specific AiActions may be triggered inside or outside of the Chat window.

MR acceptance checklist

Evaluate this MR against the MR acceptance checklist. It helps you analyze changes to reduce risks in quality, performance, reliability, security, and maintainability.

Related to #564979 (closed)

Edited by Matthias Käppler

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