Good question and your assumption as to why not is pretty spot on.
In most cases, the Drop-Ship supplier you deal with is not the manufacturer of the products they are selling. They are just a wholesaler who have made a bulk purchase and are distributing the products to resellers through a Drop-Ship supply method.
So to a Drop-Ship supplier like that, it makes no sense whatsoever going to the trouble of labelling products differently per customer's requirements, especially given that they could be dealing with hundreds of different customers.
Can you imagine the nightmare it would be trying to manage that type of setup for the supplier?
Besides that, their entire business model is to have no supply identifying labelling at all, so it's not something that is usually even on the Drop-Shippers radar at this point.
Now, when you move to the manufacturer of a product themselves who offers a Drop-Ship model direct to resellers, you still have the issue of how would they manage offering that kind of service for hundreds of resellers. You could have a situation where one reseller is moving 500 of their units a week and another reseller moving one unit every 3 months.
So just from a purely logistical perspective, it's really unmanageable unless you limit the number of resellers you offer your private label service to and in so doing, creating a premium service that is open to X amount of resellers.
Where you will find this service being offered more and more in the future (in my opinion) is among both manufacturers and wholesalers who offer their own fulfilment services. So you are essentially buying your stock in volume, they are then labelling and storing and then shipping your own brand for you.
Any wholesalers doing this would, of course, then require the manufacturer's permission to offer Private relabeling of their products.
So it is hard to imagine any supplier offering a Private Lable service, Manufacturer, Wholesaler or Drop-Shipper, that is based on a single item processing platform. Some smaller boutique suppliers may well be willing to offer it under certain circumstances, as a way to push products into the marketplace and increase their bottom line, which is the most important thing for any small player trying to get a foothold in the market ...... trust me, this is something I understand very well first hand lol because I'm in that exact position myself currently :)
Anyway, hopefully, that gives you a general overview of how difficult it would be from a supplier's perspective to offer that type of service on a single item distribution platform or at least offer it to a wide, non-exclusive, market.