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, butin 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 the position can be turned into a postdoc positionsuch conversion is possible.
Benefits:
You are now already on the PI's radar, which is a positive thing if you're a good candidate.
The PI might appreciate that you proactively sought contact. That's a good behavior pattern.
If the particular position cannot be turned into a postdoc, you will know it earlier and save time for applying and waiting.
Drawbacks:
- 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).