A set of standards has existed. The golden rule is not without assumption and cultural bias. I think that Dennis E Taylor points out the flaw in your assertion well in the novel Heaven's River. In it, an alien race has the golden rule as a cultural touchstone, and a rule in their culture that states what you said. However, the alien points out that what you have described is actually not the golden rule, but the silver rule, saying:

There are three rules of behavior...

• The Iron Rule: Treat others less powerful than yourself however you like.
• The Silver Rule: Treat others as you'd like to be treated.
• The Golden Rule: Treat others as they'd like to be treated.

The cultural assumption we make of the golden rule as you put it is that the burdens of interaction and the interpretation thereof lie squarely on our interlocutors, rather than on ourselves. Under the Silver rule, if you would not, yourself, be bothered by someone playing their music out of a speaker on public transit then, by treating others as you'd wish to be treated, you can play music loudly on public transit without doing anything wrong. There is nothing wrong with an utter lack of consideration of others under the Silver Rule. Taylor uses the example of a Carnist offering a Vegan a steak when they are hungry. The vegan would refuse, and likely be offended, while the Carnist would have done nothing morally objectionable under the Silver rule.

There is no set of rules which is perfect and unassailable, because everything we do has cultural context. This other, alien version of the "Golden Rule" comes from the assumption that, as moral creatures, it is our moral duty to communicate and determine the preferences of those around us before taking any action which could affect them. I'm sure that someone with a different cultural context could make the case that there is a flawed assumption in the alien's Golden Rule, as well. (In the book, Taylor points out that this other golden rule implicitly privileges ridiculous beliefs and preferences, potentially leading to unreasonable outcomes)

I must say I'm slightly irked by yet another vague and unfitting application of Godel's incompleteness theorems (GIT). The first one roughly states that given a consistent system F there will be some sentence GF such that you cannot derive in F that GF or not(GF). You can then add GF as an axiom to F resulting in F' but then you will be able to construct GF', and so on. This GF you can construct is called the Godel sentence of F.

If you were to encode some rules of ethics in a formal system E you could interpret it such that X is ethical (or not) if you can derive some sentence S (or not(S)) and that sentence S represents X. E will not decide on everything indeed if E is consistent (thus never deriving both some S and not(S)) and capable of basic arithmetic (what does that mean in terms of ethics?!). However, to conclude anything non-trivial from this is dangerous. First of all everyone then has to interpret what the sentences in E mean in an universal way. Then if you construct a GE does that sentence even mean anything intuitively? Will there even be any other sentence not derived from GE that is not decided upon? It is likely that E, while not deciding on everything, decides on much more than any persons ethical intuition. Surely if E is non-trivial there are some complicated sentences that it decides on that no person could interpret. And don't forget the arithmetic issue. If you just include arithmetic in E then all undecided sentences may simply be sentences of arithmetic. If the ethics part of E and arithmetic part are one whole, what would that even look like?

GIT is quite technical and applying it is troublesome. I have only read a bit of actual literature on the topic and even with my limited knowledge of it I can tell that the applications I see in the wild are shaky at best.

there”s no possible set of rules that will completely coincide with our intuitive sense of good and bad behavior in all potential scenarios.

If you just want to come to this conclusion, all you need is a counter example. Take some system of ethics E. If there is any situation such that some person A asserts S and some person B asserts not(S) then either E only decides on at most one making E not universal or E decides on both making E inconsistent. Basically if there is any disagreement on ethics (which there certainly is) you simply cannot construct a consistent and universal set of rules. We really don't need to open Pandora's box and get GIT involved.

Sorry for this wall of text. I'm not angry and I don't want to start a fight lol. I just think that people are a little too eager to apply GIT to all kinds of things. A little pet peeve of mine I guess.