Can you help me figure out the が particle usage in this subordinate clause?

I’m not following the connections you’re trying to make.

No. が as a nominaliser (i.e. like modern の) is completely dead and survives only in specific fossilised expressions like the one you mention.

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Would the same phenomenon occur with words like 上手? Am I writing this correctly?

「私は泳ぐのが上手です。」
「他に誰泳ぐの上手ですか。」
「マキさん泳ぐの上手です。」