Perhaps the word you are looking for is one of the following: "quintessential," "benchmark," or "standard."

"Quintessential" (adjective) is defined as:

Representing the most perfect or typical example of a quality or class. [Lexico]

"Benchmark" (noun) is defined as:

A standard or point of reference against which things may be compared or assessed. [Lexico]

"Standard" (noun + adjective) is defined as:

(noun) An idea or thing used as a measure, norm, or model in comparative evaluations. [Lexico]

(adjective) Used or accepted as normal or average. [Lexico]

I personally love using the word "epitome". Check out the Oxford English Dictionary's definition of it:

epitome: A person or thing that is a perfect example of a particular quality or type.

According to Merriam-Webster, there are two senses of gold standard. The first is the economic sense that isn't wanted.

The following is the second sense:

2 : BENCHMARK sense 1a

Rather than it just being one of several synonyms, it's specifically called out in the definition, which means that it's the closest of synonyms in meaning—at least according to this dictionary.

Looking at that sense of benchmark, we see this:

1 a : something that serves as a standard by which others may be measured or judged
       // a stock whose performance is a benchmark against which other stocks can be measured

The other related sub-sense that adds to the specific sense of testing in the question is this:

1 c : a standardized problem or test that serves as a basis for evaluation or comparison (as of computer system performance)

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While other synonyms are also close, it seems as if this is the one that is the closest.

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Note, however, what the Oxford (Lexico) definition says of the second sense of gold standard:

1.1 A thing of superior quality which serves as a point of reference against which other things of its type may be compared.
‘breast milk provides the gold standard by which infant feeds are measured’

Oxford's definition doesn't even reference benchmark, or any other synonym. The definition it gives is even more precise than just benchmark, since it defines it as both "a point of reference" and "of superior quality."

It doesn't really matter what its etymology is, only what its current definition and senses are.

As such, I believe it's a misconception to think that gold standard is actually the wrong word (or term) to use. Since it has a second sense, one that is far more accurate than any synonym, at least according to Oxford, I would just keep using it.

However, if not, then benchmark would still be the closest of the synonyms.

Another word that could be used here is touchstone.

Touchstone: An established standard or principle by which something is judged.

Example: Until relatively recently, the Japanese car industry was the touchstone of international success. [Cambridge English dictionary]

Exemplar could also be used.

I think industry leading techniques would work well in your examples.

You could also replace *gold standard with paradigm (formal) or paragon (though I've never heard them used in medical studies).

Paradigm (noun): A model of something, or a very clear and typical example of something.

Example: Some of these educators are hoping to produce a change in the current cultural paradigm. [Cambridge English dictionary]

Is there a better term to use in place of “gold standard”.

Probably not - it is quite a precise term.

I have to take issue with some of your assumptions:

It is common to see things described as the "gold standard". It means that something is the best and is the ideal against which other things are to be compared.

This is accurate. Gold provides a source of absolute value and thus a comparative base line for value. This derives from the pure desirability of gold that had no practical use at all and was thus not subject to market considerations, rather it was priced at a proportion of what the sum of the economy could afford to pay.

I don't like this term.

Unfortunately, one person's subjective view is unlikely to be of influence in the evolution of the language.

First, it's confusing;

It is not confusing at all. "A gold standard for X" is that standard by which all others Xs are judged.

if you Google "gold standard" you will mostly get results for the monetary concept, not the more general phrase.

This is not an argument. Google reflects what the world is discussing at any one time. Its entries are not sorted by meaning. It is not a dictionary.

Second, the etymology is broken; the monetary gold standard was effectively abandoned in the 1970s.

This is incorrect. You have given the etymology. People understand the etymology, and even if they do not (which is unlikely), they know what meaning the phrase conveys.

I often see it in the medical domain, where a certain diagnostic test ... is considered the most accurate and then other tests ... are then evaluated against this benchmark.

Which is exactly what happens with gold and money.

Words and phrases develop over the years. Specialised concepts often form attractive parallels and thus a word or phrase will extend in its meaning as a metaphor. Metaphor is well established in English.

Of the choices above, I would suggest "paradigm" or "benchmark" as a close synonym.

OED

Paradigm: 4. A conceptual or methodological model underlying the theories and practices of a science or discipline at a particular time; (hence) a generally accepted world view.

1962 T. S. Kuhn Struct. Sci. Revol. ii. 10 ‘Normal science’ means research firmly based upon one or more past scientific achievements..that some particular scientific community acknowledges..as supplying the foundation for its further practice... I..refer to [these achievements] as ‘paradigms’.

Benchmark: 2.a. In extended use. A point of reference, esp. one from which measurements may be made; something that serves as a standard by which other similar things may be measured or evaluated;

1986 J. Cox Spirit of Gardening 70 A huckleberry pie that to this day remains the benchmark against which all subsequent pies are measured.

Best practice (industry best practice, best practices, etc.). If you want to emphasize that you are trying to improve upon it, you could say the current best practice.

From Lexico.com:

commercial or professional procedures that are accepted or prescribed as being correct or most effective.

There are some healthcare specific uses, from a whole Wikipedia page of uses:

Research Validated Best Practice: A program, activity or strategy that has the highest degree of proven effectiveness supported by objective and comprehensive research and evaluation.

Especially in medical diagnostics, what is the gold standard today can be superseded by better methods tomorrow. As a matter of fact, that is what most people investigating better methods are actively trying to do.

Thus, I think a better term would be state of the art, which can be used as a noun or an adjective, just like "gold standard". This accurately conveys that we are discussing the current state, not some abstract ideal.

Merriam-Webster:

being or involving the latest methods, concepts, information, or styles

Lexico:

Belonging or relating to the most recent stage of technological development; having or using the latest techniques or equipment.

I would use quintessence. Though less common than quintessential, it's not too formal and carries a similar meaning.

A thing that is the most perfect example of its type; the most perfect embodiment of something; epitome, prototype.
— Wiktionary

In support of Paragon (already mentioned in passing): etymonline says:

"a model or pattern of special excellence or perfection; a person of supreme merit or excellence," 1540s, from Middle French paragon "a model, pattern of excellence"

However, following the thread further, it turns out to be an etymological descendant of a word meaning (basically) "touchstone" which, of course, is closely related to the controversial much-loved or much-hated "gold standard".

Mirriam-Webster includes this information directly without worrying about the French:

Noun
...derives from the Old Italian word paragone, which literally means "touchstone." A touchstone is a black stone that was formerly used to judge the purity of gold or silver. The metal was rubbed on the stone and the color of the streak it left indicated its quality. In modern English, both touchstone and paragon have come to signify a standard against which something should be judged. Ultimately, paragon comes from the Greek parakonan, meaning "to sharpen," from the prefix para- ("alongside of") and akonē, meaning "whetstone."

So it is still related to the gold standard; but it is not immediately obvious. However, it is very well-fitted to your question, as it is also a verb meaning "to compare":

Definition of paragon (Entry 2 of 2)
transitive verb
1 : to compare with : parallel
2 : to put in rivalry : match
3 obsolete : surpass