I'm not a historian by trade, but a few years ago I spent an inordinate amount of time reading primary source material about the whole Reynolds Affair. The short answer to your question is, "No. It was never provien that Hamilton used public funds to pay off James Reynolds". And trust me, many Anti-Federalist congressman would have been thrilled to have been able to do just that.

The longer answer is that so many aspects of the Reynolds Affair are still cloaked in mystery and will never be conclusively "solved". Of all my reading, the most concise and academically rigorous of all the summaries of this affair comes in an "Introductory Note" contained in the long project of collecting the writings of Alexander Hamilton. It can be found here:

https://founders.archives.gov/documents/Hamilton/01-21-02-0076-0001

This particular introductory note was written many years ago by Barbara Chernow (just a coincidence, no relation) while working on the project at Columbia under the leadership of Harold Coffin Syrett, the Executive Editor.

Sadly, we have very little primary sources available to us about what happened during all the twists and turns involved in the Reynolds Affair beyond what Alexander Hamilton himself wrote in the later pamphlet or what he claimed to have had in terms of original letters, etc. There are no primary sources from the perspective of either James Reynolds or Maria Reynolds, and only the barest of snippets from Jacob Clingman, who was wrapped up with James Reynolds in some illegal schemes.

I am of the of the opinion that it is exceedingly difficult to understand the nuances of a society that is so foreign to modern sensibilities. 1790s Philadelphia is light years removed from today, and thus it is dangerous to believe we understand what was going on. It certainly makes for a fun puzzle though. It's like seeing 10 or 11 pieces of a thousand piece jigsaw puzzle of a picture you can't even see and trying to parse the particulars.

One interesting side note, however, is that a famed UVA historian was convinced that Hamilton had plagerised the letters he claimed had been sent to him from Maria Reynolds. There is only circumstantial evidence of that claim, however, because the originals of those letters simply don't exist. All we have is Hamilton's transcriptions of the letters he said were from Maria. Anyway, that's a whole other story, and involves a tangential character who was a dwarf printer in Philadelphia.