Since I wasn't quite convinced by what I'd seen so far, I decided to do some more digging. This goes down through multiple layers of research:
There are two ways to place emphasis on the words "gold standard", and each implicitly indicates a subtly different meaning. When people talk about the economic policy, they usually say "the gold standard" (emphasis mine for exaggeration), i.e. "the standard of [using] gold". But when I hear people use the phrase in a medical/scientific context, I've invariably heard them stress it as "the gold standard", i.e., "the golden standard" (and I work with statisticians and public health professionals, so I hear that phrase used a lot). It's the difference between saying "the gold egg" to mean an egg that somehow hatches gold, vs an egg made out of gold - the difference gets coded into how you say it.
So there's clearly (to me, at least) an implicit consensus among the people using the phrase that (when used in a scientific context) they're talking about a golden standard. But this doesn't rule out the possibility that "gold[en] standard" is an egg corn. People may well have gotten confused as to the origin of the phrase, since (being used largely in technical papers and scientific publications) most people are going to first see this phrase written down rather than spoken.
Many standard dictionaries, such as Collins' English dictionary, define the phrase in two ways - the economic way, and the scientific way - side-by-side. It's not unreasonable to think that this suggests the one usage led to the other. But it's not ironclad either: convergent evolution of language is not unheard of.
This site aggregates definitions from a variety of medical dictionaries. Two entries stand out as interesting. First, the Farlex Partner Medical dictionary notes that
The term criterion standard is preferred in medical writing.
I read this as saying that "gold" is an informal and jargonistic way of saying "best" standard, and they prefer a more technical (but gramatically equivalent) phrasing. Stronger evidence comes from the entry in Saunders Comprehensive Veternary Dictionary, which defines it as
The ultimate standard to which all endeavors aspire.
Once again, this suggests that "gold" is a stand-in for "best" or "ultimate" rather than referring (directly, at least) to the metal.
History of the phrase
Fortunately, back in 2005 there was an article published in the British Medical Journal trying to answer exactly this question. There are several highlights. First, it notes that the common usage does indeed assume "gold" meaning best:
“Gold standard” is the popular term to describe this test; but “golden standard” is sometimes used as well. In fact, almost all medical publications in Dutch use the term “gouden standaard” which is a translation of “golden standard.”
The author argues that this is a mistake:
Inspired by the Olympic Games, where the best athlete wins the gold medal, people who use “golden standard” think the term denotes the best standard in the world. Not bronze, not silver, but gold. Of course, this is incorrect.
In addition to quoting a paper from 1992 that compares its usage in medical science to the old economic gold standard (which suggests that at least some professionals who use the term think of it as relating to monetary policy), he also traces its origin in medical science. It first appeared in reference to "set[ting] a standard for the use of gold salts in patients with rheaumatoid arthritis", and its first appearance of the phrase in its modern (scientific) usage is a paper written in 1979 entitled "In search of the gold standard for compliance measurement".
Case closed, right?
Original appearance of phrase
Most of the 1979 journal article is behind a paywall, but fortunately the first page already contains the author's first use of "gold standard" (emphasis mine):
While other methodological problems are involved, these four highlight the absence of a real gold standard for compliance. The ideal standard would be...
Once again, we see the usage of "gold standard" interchangably with "ideal standard". While this muddies the waters a little, the other appearances of "gold standard" in that article do seem to line up with it being a reference to the gold standard for currency. And attempting to read any further into this article is likely fruitless.
The BMJ article comes down pretty hard on the side of "the gold standard"'s origin being a reference to monetary policy - and the sources it cites, for the most part, support that interpretation. But it's also clear that the amgiguous usage - and the possibility for inerpretting the egg corn of "golden standard" from it - has been present from the start. It's not hard to imagine that the alternate interpretation has existed almost as long as "the gold standard" has been used in medical science. I suppose there's no way to really definitively settle the question short of calling up Peter Rudd and asking him what he meant in that original article, but barring that it seems like the BMJ article at least brings us closer to understanding the history of the phrase.