“Chareidim may think they’re getting a Badatz hechsher, but in reality, are getting the regular Rabbunut.”
From what I understand, the difference between a Badatz (meaning Chareidi) hashgacha and a Rabbanut hashgacha does not necessarily have to do with the Mashgiach himself but rather with the standards that the place is required to comply with in order to receive the hashgacha. In other words, the same mashgiach could be willing to give the same place a Rabbanut hashgacha but not a Chareidi hashgacha if their standards make them eligible for one but not the other.
You can see this from the fact that there is such a thing as Rabbanut Yerushalayim and Rabbanut Yerushalayim L’Mehadrin. I assume the mashgichim are the same (or could be), but there are different standards.
When I have had to attend Simchas that had a Rabbanut hashgacha or had to eat out at Rabbunut restaurants with relatives, people have told me that I could ask the mashgiach what foods I could eat. Personally, that seemed strange to me (and I never checked that out with someone I trust), so I just stuck to good old Coca Cola or Pepsi Cola (but not the diet since that had herbs or something floating in it to differentiate it from the regular soda).