The pull request that @QuietMisdreavus made to enable that was merged 3 days ago (Feb 28th), so it definitely won't be reflected in the swift versions that are available stock from Xcode, and probably not included in the Swift 5.8 final release.
If you were to get a nightly toolchain, you might be able to make it work with that - as there was one from March 2nd that's available - but I'm not sure how the Xcode project parameters flow through, and if that's a feasible solution for you at the moment.
The nightly toolchain with my PR will be able to accept -skip-protocol-implementations in OTHER_SWIFT_FLAGS just fine. The reason that -skip-synthesized-symbols doesn't work is that it wasn't set up to be accepted during a regular compilation, just the swift-symbolgraph-extract tool. However, since Xcode doesn't run that tool, it doesn't work. 
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Am I able to run that tool manually by chance? I noticed it was a symbol graph only flag but just not sure exactly what that means. Also thanks for all the insight!