I would suspect the reason why you’ve never heard a Federation Citizen explain why Intaki has the right to secession could very well be explained by that it has never come up in conversation with any that you have encountered, or that you have never asked them.
It is possible that the average Federation Citizen may not be aware of the Intaki Secessionist movement, considering that they have their own concerns and politics at the forefront of their mind within the confines of their member state, likely unaware of what goes on outside of it, or their District. Strewth, many are not even aware that a Federated Union even exists, depending on the area where they live, until the time comes to elect their Senatorial representative or Presidential Candidate.
Of course, when we speak of Intaki, are we referring to the Intaki Solar System, the District, the Intaki Homeworld or those that identify as Intaki or having Intaki ancestry, numbering almost a quarter of the Federal population? Intaki could mean any of those and more, it isn’t a word that is defined by one meaning when the topic of secessionism or independence arises on these forums or in political discourse.
For simplicities sake, and the Federation is anything but simple when delving into it’s intricacies, I shall presume that you are referring to the Intaki Liberation Front’s stated goal of the independence of the Intaki system. The honest answer is, there is not anything preventing a member state from declaring independence from the Federation in theory or in practice. The Federation is an interstellar alliance based in part on the principle of membership and acceptance by mutual consent, not one of conquest or subjugation. I am not aware of any examples of legitimate separation from the Federation (beyond the obvious example, although the legitimacy of it is questionable depending on whom you ask).
As much of an idealist that I am in that I believe that those member states in the Federation have come to adopt the ideals that it espouses out of righteousness, I accept that there is a very realist argument to accepting the Federation’s laws when signing the Charter and agreeing to be bound by the Constitution and Federal Law. One of them is that it is an insurance guarantee of the member state’s sovereignty, as the Federation is bound by contract to render aid and assistance should it be attacked. An attack on one truly is an attack on all, regardless of whether it is the richest core member state or a bootstrap colony on the fringes of Federal space. The economic benefits, development opportunities, freedom of movement, mutual co-operation and the defence of each member state’s way of life and culture; all make membership attractive.
The real question to be asked is: what does a signatory gain by withdrawing from the Union? Sovereignty or ‘regaining control’ over one’s own affairs sounds like an attractive buzzphrase to be used, but the end result of withdrawal would more often that not likely to be detrimental more than beneficial. And quite frankly? The Federation does not nearly have much in the way of direct rule over each member state, contrary to what you may think. Aside from abiding by what obligations they have to as part of the Federation (which could be minimal, if opt-out clauses come into play which many Intaki member states have taken advantage of), it’s presence is barely if at all noticeable.