When we teach private OW classes, all our students are in backplate setups with primary donate.
There is a LOT of nonsense in scuba shops, though. You do need to check with the shop or instructor where you intend to do your OW, and make sure they/he are OK with your gear. You may get resistance. You may be told they want all the students in the same gear (theirs) or that you have to get your gear "serviced" before they will approve its use. They may balk outright, being unfamiliar with the equipment and resistant to doing something different. Shops vary in their flexibility and willingness to meet the customer's requests.
There is a LOT of nonsense in scuba shops, though. You do need to check with the shop or instructor where you intend to do your OW, and make sure they/he are OK with your gear. You may get resistance. You may be told they want all the students in the same gear (theirs) or that you have to get your gear "serviced" before they will approve its use. They may balk outright, being unfamiliar with the equipment and resistant to doing something different. Shops vary in their flexibility and willingness to meet the customer's requests.