You're talking about a country where "I want a lawyer, dawg" is not sufficient to be provided a lawyer, because the cops can reasonably think that you want a dog that's a lawyer. The legal system in the US is beyond saving.

Because the US legislature stopped making laws, courts make laws instead, and they do it based on who's taking them on luxury trips.

The US legislature is in effect half a dozen sugar babies.

You're not wrong.

There is a rationale, it just fails to be consistent with the reason the right is explicitly enumerated.

Some rights are more about restraining the governments actions towards you. The right to legal counsel prohibits the government from putting you on trial totally lost and oblivious. The right to remain silent is a nickname we give to the prohibition on someone being "compelled in any criminal case to be a witness against himself".
Rights that prohibit government actions are usually automatic: free speech, no unreasonable search and seizure, and so on all prohibit the government from doing something.
The right to counsel requires the government to provide you with a lawyer if you need one, and to stop asking you questions if you ask for one until they show up.

The conservative supreme Court majority held that the "right to remain silent" was a right to make the government stop, just like asking for a lawyer, and that it didn't follow that the two rights needed to be invoked in different manners: no one has argued that their behavior should have implied they wanted a lawyer even if they didn't say so. Further, they said that if "not speaking" is what constitutes invoking the fifth, then it opens more questions about what manner of not speaking counts as an invocation. If someone doesn't answer, can the cop repeat themselves?
Invoking the right however is unambiguous, making it a better legal standard.

This falls apart not because of the nickname we give the right, but because of what it says and the intent behind it: "No person ... shall be compelled in any criminal case to be a witness against himself". The key word is "compelled". The phrasing and intent are clear that it's about compulsion , not invoking an entitlement. Any statement you make you should be able to refuse to allow to be used against you, otherwise it's compulsion even if given willingly at the time.

Remember, the system of "law" was never designed to help the people, only reinforce the rule of the owning class over the masses. The system originates from the dictates of kings.

The modern iteration is intentionally convoluted to ensure that the majority, who cannot afford quality counsel, cannot leverage the law against the owning-class, who have access to insurmountable resources and legal advantages.