Your leadership is technically in the right. You cannot have more than one primary persona. It sounds like you have split your personas too readily. If you find that two personas seem to vary only slightly in motivation, but not behaviour, you may choose to eliminate one of the redundant personas or tweak the characteristics of your personas to make them more distinct. If that's not the case, you'll have to designate some persona's as secondary or even supplementary.
The book About Face sums it up quite nicely:
Each persona must vary from all others in at least one significant behavior. By making sure that your persona set is complete and that each persona is meaningfully distinct, you ensure that your personas sufficiently represent the diversity of behaviors.
You'll need a proper set of persona's in order to design. But you'll also need to choose which users to prioritise. You'll need to meet the needs of the most important users, without compromising the ability to meet the needs of secondary users.
Primary personas represent the primary target for the design of an interface. There can be only one primary persona per interface for a product, but it is possible for some products (especially enterprise products) to have multiple distinct interfaces, each targeted at a distinct primary persona.
A primary persona will not be satisfied by a design targeted at any other persona in the set. However, if the primary persona is the target, all other personas will not, at least, be dissatisfied.
Choosing the primary persona is a process of elimination: You need to compare goals of each persona against the goals of the others. If you cannot find a primary persona, it means that either your product needs multiple interfaces, or your product is trying to accomplish too much. Your scope might be too broad.