Under which circumstances (if ever) should a prospective post-doc apply for a position advertised for a PhD student?

It does not make any sense, at least in Europe. Do not know, maybe it is different somewhere else. Do not waste the paper on such an application, protect the planet.

While it is possible to have a very naive view that more competence and experience always makes the better candidate, scientific world also has many other restrictions directed against "professional PhD students". I remember trying this really hard out of desperation, many years ago. At that time I already had post doctoral positions in leading universities, publications in good journals, but eventually was forced to yield and go to the industry, like everybody does. At the same time, people right from my university (same master degree, same specialty, very comparable grades) where finding PhD positions no problem.

The only way how it could possibly work is if the professor has multiple options and can convert a PhD grant into something that suits for you. But I have never seen this happening.

Things are going to differ country to country, field to field and situation to situation but often PhD studentships are funded by outside funding bodies (this is not the case in the US interestingly). Those funding bodies are not primarily interested in the research that is produced from a PhD project, but in the education provided. All those PhD students that don't go on to become Postdocs? Many of them go and use their PhD in industry, which is an economic boost.

Most PIs I know would indeed rather have a postdoc than a student, but in the end its not the PI that controls the money or the priorities, its whoever is providing the funding.

That said sometimes a PI might have a non-externally funded position, or funding from a more fungible source. Normally they would aleady be advertising a postdoc positions, but I have known a PI convert 3 years of a PhD student into 1 year of postdoc (postdocs cost 3x as much as PhDs here) before.

While it might sometimes be possible to convert the funding from a PhD to a postdoc position, in general I don't think it's a good idea to apply as a postdoc for a PhD position. That being said, you can always contact the PI and ask whether they would have another position for a postdoc.

• Often, there are many more projects advertised for prospective PhD students than for prospective post-docs.

This is due to several reasons:

• First not every PhD student continues as a postdoc: a lot of them go to industry, some of them quit or fail... and some of them achieve a mediocre PhD which doesn't allow them to get a postdoc job. So mathematically there must be more PhD positions than postdoc positions: if say only 20% of PhDs continue as postdocs, there must be roughly 5 times more PhD positions offered than postdoc ones.
• Universities have a duty to teach students, they don't have a duty to hire temporary research staff. The number of PhD students taught is an important target indicator for an institution, whereas the number of postdocs is more a result of their success at grant applications.

• Many of the projects seem to be scientifically challenging enough to be interesting also for a post-doc, especially if one is switching fields or completing the project in a shorter time frame.
• I cannot see why institutions would be opposed to hiring post-docs instead of PhD students, considering the much higher qualification they bring. I can imagine that there are some limitations attached to funding, though.

I've heard the very vague estimation that one year of postdoc is equivalent to 3 years of PhD in terms of research productivity (incidentally, they are often paid around 3 times more in countries where the PhD is a grant). There's some truth to the idea that a postdoc can do the same job as a PhD faster. However there are also important differences which can make a PI choose a PhD rather than a postdoc:

• a PhD topic can be more prospective and evolve over time.
• the PI might want the PhD to test their own research ideas, so they want to supervise the work closely. On the contrary, a postdoc is usually more independent.
• supervising PhDs is a must to advance their career.
• co-supervising a PhD student is a common way to start a collaboration with a colleague, whereas a postdoc doesn't need much supervision (if any).

It's important to understand that universities are not like companies, they are not looking for the best quality/price ratio for a task. The goal of funding a PhD is not only to answer a research question, it's also to teach somebody how to do research so that they can contribute to society later. This is why the main reason for an institution to hire PhD students is simply that it's an essential part of their mission, that's why they receive public money. This is also why there are indeed very often limitations on the funding, simply because a lot of PhD funding comes from national or international programs meant to increase research capacity by training researchers.

Does it make sense to apply for such a position even if you already have a PhD, in the hope that they might also consider a post-doc instead?

Yes, in general, this makes sense, because there is a chance that the position can be converted into a postdoc position. Whether that's indeed the case will depend on the nature of the funding. In some cases it will be possible, in others not.

Therefore, before you apply, it's best to contact the PI and ask them if such conversion is possible.

Benefits:

1. You are now already on the PI's radar, which is a positive thing if you're a good candidate.

2. The PI might appreciate that you proactively sought contact. That's a good behavior pattern.

3. If the particular position cannot be turned into a postdoc, you will know it earlier and save time for applying and waiting.

Drawbacks:

1. Apparently none.

Why would the hiring institution not want a post-doc instead of a PhD student?

There can be rules, for example, if dedicated funding is allocated for PhD students. I know that this is the case for some positions in Germany. Beyond rules, there can be some benefits in hiring a PhD student. For example, graduated PhD students is a success metric in hiring decisions.

Why would a prospective post-doc not want such a position?

From the perspective of the post-doc, I don't see how the position (if a conversion is possible) would be different than any other post-doc position. This assumes that the institution not tries to lowball your salary somehow because the position was initially meant for a PhD student (which would be a red flag).

General answer: Where there's a will there's a way. If the PI/lab is interested in your talent or proposal, there is a chance they will make it work.

I think you are correct noticing that advertisement for position means they need something. And you as postdoc can fill the gap. The issues is that PhD positions often seems to be funded from different sources than postdoc positions.

That being said, you can only try to reach out. There is nothing you can lose.