Here in the USA, we have numerous substantive and procedural criticisms of the legal system, and while IANAL, the latter is of particular interest to me and is the in domain of your questions. I will try to address each in turn, since they kinda build upon each other.

Shouldn’t [providing a lawyer] be the default and not require the suspect/subject to actually ask for one?

To get to the answer, we need to step back and get examine what the exact obligation is. In the USA, the specific right in question is the individual’s right to choose legal counsel. That is, a person has the final authority as to who will represent and advise them in legal proceedings. This right isn’t unlimited though, and it doesn’t mean that they ought to be represented by a specific lawyer for free. But rather, the right means that no one else can make that decision on that person’s behalf.

But in the Anglo-American formulation of what a right is, it is also an obligation upon everyone else. Specifically, the government is obligated to not interfere with a person’s free choice of lawyer. This was poignantly and recently examined by the federal court in DC, as it pertains to the executive’s attacks on the law firm Perkins Coie, where the federal judge ripped the government for interference with due process rights, from which the right to choice of lawyer.

But there’s a wrinkle with rights: if the liberty it affords is the ability to choose, how would choosing nothing be handled? That is, if a person wishes to not choose, how can they affirmatively decline to choose? There are – and it’s a foolhardy exercise – criminal defendants in the USA that plainly choose to represent themselves in court, with no lawyer to aid them. The general rule for a “unilateral” right such as this one is that it is “optional”, where affirmative actions are needed to involve the right, otherwise the default is that the right isn’t invoked.

And that sits fairly well in the breath of rights that civilians enjoy, such as the right to travel the public lands (eg walking or riding a bicycle on the street) to the First Amendment’s right to petition the government. After all, no one from the govt is phoning people up every day to ask “do you wish to unicycle on Main St today?” or “would you like to comment on the city budget next Tuesday?”. More clearly, those rights are fairly obvious when they wish to be used, or when they don’t wish to be used. (Though I grant you that the latter implicates a right to notification, but that’s a whole different matter)

The system of rights gets even more complicated when someone holds two opposing rights. For example, in the USA, everyone has both the right to free speech, plus the right to silence. In that case, it absolutely forces the matter, because the absence of speech is very much a matter than can be criminalized. For example, failing to say something under penalty of perjury. How this is handled gets complicated, and generally speaking, such actions or inactions have to clearly show intent to invoke (or not) the specific right. This is precisely why it’s important to say “I wish to invoke my right to silence and to an attorney” when arrested, because otherwise the government’s obligations are confused, since the rights are confused. That statement unquestionably clears up the situation for how the govt must behave.

Basically, in order for the govt to meet its obligation not to interfere with someone’s choice of lawyer, it would not be proper if they then proposed a lawyer by name to represent that person. Even just making such a proposal is coercive, since the govt holds most of the power and clout when in court. Instead, if the person voices their request for a lawyer, then that sets into motion the apparatus for verifying their eligibility for a public lawyer from the Public Defender’s office – btw, these offices are woefully underfunded, so contact your representatives to fix this! – and then finding such a lawyer to represent the person.

All of this stems from due process, and the “Miranda warning” is the practical implementation of due process. Since if someone doesn’t even know they have a right, it might as well not exist.

I think the only question should be “do you have your own lawyer you like to use, or are you happy enough with the court-appointed one?”

This is the obvious question, following notification that the right even exists. But again, if the appointed lawyer has already been selected and it’s only a trinary choice - your own lawyer, this specific public defender, or no one – then that’s still somewhat coercive. It precludes the possibility of having a different public defense lawyer, of which the existing process already handles.

When I say that the public defender’s office finds a lawyer to represent someone, they do so while mindful that not every lawyer can represent every client. After all, Greenpeace wouldn’t want a lawyer that’s also currently working a case for Chevron, the oil giant. Conflicts of interest may arise, as well as any other scenario that would make said lawyer less effective at their job: zealously advocating for their client.

But again, this isn’t an unlimited right of the person, so a case cannot be delayed indefinitely because the client doesn’t like any of the public defender lawyers. But a case can absolutely be parked due to no available public lawyers, though if this happens, courts typically have other avenues to clean the logjam but without infringing on civil rights.

Has there ever been any attempt to make that the norm in any countries?

I’m only vaguely familiar with Anglosphere jurisdictions, and haven’t come across a system that improves on this situation. Though quite frankly, if it’s going to happen, it should be tried at the state level in the USA, where there’s the most room and latitude for improvement.

I’m not even sure opting out should be allowed, but I’m open to hearing reasons why that would be a bad system

The coercion issue from earlier can be turned to 11, if the govt is operating in bad faith. Imagine, for example, that the govt charges someone with bogus accusations, then bribes a corrupt lawyer from out-of-state to come represent the defendant against their will, who will then “throw” the case and land the defendant in prison. There are a lot of norms and procedures that would have to be violated to do this, but that’s kinda the point: defense in depth is equally applicable to computer security as it is to civil rights.

An institution that assumes good faith govt will be hard pressed to deal with a govt that acts in bad faith. I make no excuses for the numerous American federal and state-level judicial fails, but when it comes to institutions that will uphold civil rights, individual liberty with regards to accessing the legal system is crucial.