> 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, but before you apply, it's best to contact the PI and ask them if the position can be turned into a postdoc position.

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).