Seems to me he addressed your question the very paragraph you quoted. They have the magical oomph. The game represents this in multiple ways, not the least of which is full casting ability.
There are always possible counterarguments: Clerics and Druids have full casting ability, but cannot use a Staff of the Magi; Wizards understand spells through hard study, while Warlocks just get them as part of a deal and never truly understand them, so they don't have the "oomph" to also understand how a Staff of the Magi works.
Mine was of course a rhetorical question. I do not care what you or the other poster answer to "Why?". The possible answers in favor or against (including my own counterarguments, which are made up on the spot, I don't actually believe in them) are all reasonable and worthless at the same time.
A Staff of the Magi (SotM) is simply usable by anyone you want it to be able to.
There is no fundamental reason why a game where all (and only) Wizards, Sorcerers and Warlock can use a SotM is a better game or a more reasonable game than another where only all Wizards can, or another where all spellcasters can, or another where "only arcane spellcasters of at least 5th level can, provided they are Lawful Good and born on Monday", or another where only a few individuals chosen by fate can.
The only obvious discriminant is if the SotM works in a way that is not applicable to some characters. If the SotM description says something about spell slots, and the character does not have spell slots, then the SotM as written cannot be used by that character.
The rest is up to the designers, they can do what they want: they can make the SotM usable by all Mages or by Wizards only. They are both "correct design". Then if you want to play by the book you follow their decision, but you can always change it to fit your preferences and nobody has the right to tell you that you should not.