The best answer I can think of for the overall question is: That's simply not how Kant works. Let me explain why.
Implicitly, the answer is already given in Kant's universalization explained: How does one universalize a thing?
Like in the answer, the first step will also be to identify the maxim. Why is that so? You referrred to the Formula of Natural Law in comments, which is:
so act as if the maxim of your action were to become by your will a UNIVERSAL LAW OF NATURE (Ak. 4:421)
Apart from this not being the formula Kant uses in his political philosophy - and he DOES use the Categorical Imperative in context of political philosophy (see Perpetual Peace, Ak. 8:377 for the formula he uses and Ak. 8:370-80 for the argument why it is/should be THE principle in politics) - I will stick to this for the moment as the argument will essentially be the same. So, it says "so act as if the maxim ...", which means it is the maxim of an action which is to be universalised.
But what is a maxim? As my answer in the linked question points out (including sources), a maxim contains the following:
- A particular situation
- A particular intent/end [Zweck] the action is aiming for
- The particular means/action for achieving the intended outcome (and thus what Kant calls "practical rules")
I will try to make up an actual maxim corresponding to your example of leadership: If I want to envision/enact a political system [both situation and intent], there should be one person within the system who takes the lead role [means] in order to make it functional [intent].
This leaves open how the person for the lead role is found, WHO THIS PERSON IS, the legal and institutional framework that both defines the role and its limits, etc. But whatever, let us try to universalise. The principle says that every person in the same situation should be able to will this maxim as if it were a natural law, i.e. with absolute necessity and without exception.
The universalisation will end up in stating: Whenever a person wants to envision/enact a functional political system, this person should always and necessarily do that in a way so that they want that one person within that system takes the lead role.
This can be argued as false, especially for smaller communities. On the scale of society and institutionally constituted states, I think this actually holds true and is without contradiction.
The error in your thinking probably is that there are actual persons that think differently and therefore it is wrong, as some really think they should be the leader. But this, again, is not how Kant works. The universalisation is appealing to the faculty of reason of every (finite) rational being, not the beliefs of a particular person. So in a way, it states "act so that the maxim of your action could be enacted by everyone every time they get into the situation without contradiction".
To get to the outcome you mentioned in the OP through Kantian universalisation, the maxim would have to be "If I want to envision/enact a (functional) political system, I should be the leader in that system". The universalisation of this would indeed be "Whenever a person wants to envision/enact a political system, they should always and necessarily want to be the leader of that system." This would lead to many, many one-person-states if taken seriously, everyone would necessarily want to be the leader and nobody would support any leader but themselves. But one-person-states wouldn't be political systems in any legitimate sense and hence a contradiction in conception as it lies in the very concept of a political system that there has to be a political sphere of persons in mutual relations.