This is probably very context-dependent. I have supervised PhD students at a top British university and 2 European research institutes, and usually the students had some freedom within the bounds of the project they chose initially. That’s how it should be: you want students to develop some initiative and they are not here just to follow instructions.

Now, sure it is not absolute freedom because it is supervised and the funding was obtained for a specific project. Funding agencies or industrial sponsors don’t like it when the student does something completely different. But then students apply for specific projects. I have never seen any PhD offer that did not have a set perimeter (within which they do have some freedom), so I am not sure they are being lied to about that. Again, that might be context-dependent.

Exactly, it's not about being literally unconstrained, but a lot of biology PhD programs turn into "follow my advisor's weekly (sometimes even daily) instructions to the letter". This can be very demoralizing and it is concerning for the development of future PIs as well.

In addition to micro-management of the thesis project, there's also consideration of outside academic activities. Many students end up spending 50+ hours per week on the work their advisor designed. This leaves little time for the student to explore learning new skills that might be relevant to them - for example programming tutorials, a reading group in a different subfield, mentoring an undergrad project, communicating with the general public, etc.

Obviously there is not time for every PhD to involve all of those things, but currently it is much too restricted. For some people to the point that their advisor forbids them from taking a night class or whatever despite the fact it is outside the (already overtime) working hours of the lab. Research may indeed be the priority, but entirely ignoring other parts of the academic role has not gotten us anywhere good.

What it boils down to is that part of the compensation of the PhD is supposed to be training. Yet as it stands many students are treated as only employees, with no regard to teaching them to become a better researcher and no time allowed for their own personal/career development activities.

Students are being lied to for sure (at least in certain fields/locations). But the solution is not to just be blatantly honest about what PhD currently entails. Part of the shift also needs to be a return to considering the PhD as an important piece of the training for becoming an academic.

Maybe it was somewhat unique, but when I received my offer to become a PhD student at EMBL in 1986, I wrote my own PhD proposal that was completely unrelated to any work that my supervisor was doing, and only very loosely connected to the work that anyone else was doing at EMBL at that time.

However, the funding of EMBL works fairly differently from most universities, and quite a few other research institutes.

IMHO this is the way to go. Students like these tend to have freedom to choose whatever advisor they want. For example, in Cambridge self-funded students typically do rotations to see how labs look like.

This minimizes the possibility of ending up with a bully or someone incompetent. And if you do, it's much easier to run away to a different group.

The funding makes a huge difference. A student that the advisor doesn't have to fund that has initiative and workable ideas will only require occasional face time. To top it off, the ones I've seen that were like this were usually more senior, coming from some experience usually in the industry, and as a result more savvy; therefore they knew what leverage they had and were able to walk away from bad situations. But an RA that gets paid out of grant money will have a lot less leverage.

I am a current PhD student in a CS-related field. My advisors had strict requirements for my first project, but now on my second project I have complete freedom. This was my personal choice, my advisors were/are willing to give me projects, but I wanted to learn how to take a research project from idea to finish. Obviously, CS (and my school) is extremely well funded, so this may be unique.

I think CS has pretty different dynamics than most fields for a few reasons. CS is really well positioned when it comes to both funding received and average funding required to do a project. Similarly, the time required for set up of CS work can often be minimized in a way that is not possible for many natural sciences. And culturally people seem less attached to old school academic status - CS has many viable and relevant career paths for someone that wants to exit academia, some of which even allow for fairly easy reentry to a university position.