From your question, it looks like you're thinking about it this way: that there's the underlying diatonic scale, and then the 7 modes are derived from starting at different points on that, giving us Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian and Locrian. You're then asking why only two of those modes have been given special names ('Major') and ('Minor') with the 'right' to denote keys, which is a perfectly reasonable thing to wonder!
It seems that one relevant thing to bear in mind is the way the word 'mode' has been used to refer to lots of different things over the years - The Wikipedia article on modes makes that clear, as do answers on another question from Robert Fink and David Utzinger.
Christopher Whitt highlights the point at which the idea of major and minor came into the mix:
The next milepost in the misnaming of church modes happened over a 75 to 100 year period ending in roughly 1675 ... when the church modes of Gregory were expressed as permutations of the then new major-minor scale system. That's when the modes became formalized into what we know and use today. The Greek names became convenient labels for particular scales, though there is no certain tie between the notes in any modern church mode and the notes in any ancient Greek mode. Locrian mode was 'invented' to complete a theoretical picture.
Caleb Hines has written an informative question (and answer!) on how the major / minor system slowly came to be established. once it was, it seems that the word 'mode' was then re-used yet again to mean another new thing - 'mode' of the major scale. So that seems to be why in the current, modern definition of mode, the Major/Minor system is seen as the 'parent' concept, and the modern modes are seen as a secondary concept that's derived from that.
However, in the past, the word 'mode' has been used to refer to different concepts that were perhaps more on the same primary level of importance as our modern 'keys' - some of which themselves fed into the development of the major/minor system.
Music is full of terms that mean more than one thing in different contexts - sometimes contexts do make that clear, but some musical ideas just seem to be problematic, and the idea of 'modes' seems to be one of those. Again from Caleb's answer:
Throughout the Baroque, there seems to have been widespread confusion, even among musicians, over the exact definition of a mode.
...and this gem from the Wikipedia answer:
The understanding of mode today does often not reflect that it is made of different concepts which cannot fit altogether.