Would it not be possible for one of the dynasties in charge to, in the efforts to centralize, decide to choose a vernacular tongue rather than Arabic as the language of administration?
It's really hard to pull : as mentioned above, not only medieval Arabo-Islamic power used Arab as a key cultural feature on a general basis, but al-Andalus ethnical/political features made it even more critical. One simply couldn't hope for any public role without being troughly arabized (and by that, I don't just mean converted, as muladi had to go trough a series of revolts before being admitted on higher levels).
Ottomans and Iranic empires (and in Persia, it took some times) managed to do otherwise, for two main reasons IMO : first, these empires were forged on non-Arabic (if arabized) peoples, then built on already present imperial foundations (Ottoman on Byzantine, for instance).
al-Andalus were, on the contrary, the creation of an Arab-dominated elite, which couldn't take on foundations of Gothic Spain (for various reasons, mostly being too different from the tribal/imperial features, and because they were too close of what main opponents of Arabo-Andalusians practiced).
The public and social pressure on Arabisation was really strong : Mozarabs went quickly unfamiliar with their own formal language, which is latin, at least by the Xth century.
Which lets Berber and Arab, and the first was a big no-no.
Al Andalus as a state always seemed quite internally fragile to me, relying too much on Mamluk soldiers and nobles with large landed estates rather than an internal powerbase headed by the crown.
While the really important reliance on mercenaries and foreign forces was a problem*, it eventually goes down to the political issues of the peninsula.
You had militias, but they usually played a really local role not unlike how Saqaliba could be as well tied to non-emiral/caliphal dynasties (in fact, it became more the opposite, these preferring using Berber or Christian troops eventually).
And I think part of the confusion is highlighted by your use of "headed by the crown"**.
That's too close of what one could think about feudal Europe, for that the discrepancy between both situations couldn't be highlighted.
Contrary to what existed even in the most divided feudal kingdoms, where
in fine, the "feudal contract" gave huge legitimacy to the king (and why even dynasties that looked feeble at first managed to raise quickly), it didn't really that existed in al-Andalus.
Sure, the "ultimate" ruler was acknowledged as such (most of times) but his legitimacy didn't came from its relationship to his subordinates, that didn't that owed him their position except to what matters the "enlarged" house (domestics, ministers, slaves, armies, etc.).
His legitimacy, in al-Andalus that is, was to fight Christians on a regular basis (which meant either campaigning, or more generally raiding them), but not from having handled down territories and titles to other, smaller, dynasties.
When the ruler is strong, able to crush revolts or tentative of independence***, it's not big deal. When he's not, however, and critically if it encounter a general crisis (as the latter part of the Emiral period is a mix of political, dynastic, economical and military crisis), it can goes easily to hell : simply said, you didn't have that of an obvious and strong connection between local dynasties and polities and the emiral/caliphal's.
(And it becomes painfully obvious on how taifas and local elites related to Berber dynasties after the fall of Umayyads).
You might say that the ruling dynasty could have used their position to strengthen their role, and make a similar evolution to what happened later in Western Europe...But they did, it's not like they sit on their hands doing nothing.
But it was received...poorly.
I found, some time ago, a really interesting extract on this, allow me to repost it.
It was written by Ibn Hazm al-Andalusi, in the early Taifa period.
If I make a difference between our times and previous times, it's that the exactions made during the time of the truce weren't generalized and institutionalized as nowadays, and that the conquest's tributes were made only on land property. They didn't differed much from the ones imposed by Umar.
But today, we have now : a capitation taxe on Muslims, called the qati, received each month; a tax on their goods on sheep, cows, oxen and bees consisting as a determined amount by cattle head and milking female; commercial taxes perceived on everything sold in market including the wine selling license accorded to some Muslims in some regions.
All of this, all that is covered by modern despots.
It's an abject outrage, a violation of Islamic laws, a shattering piece by piece of the commune society, the creation of a new religion when the only power in this regards is God's only! By God! If they learned that in the worship of the cross, there was something to make business, they would quickly convert to Christianity."
Roughly, the Emir/Caliph was supposed to get its fund from where it get its legitimacy : on Christians.****
It's interestingly not too dissimilar to the late medieval view that princes should take their funds during peace on their own lands, not trough taxation (which was viewed as exceptionnal and for matters of war).
But there, it collided significantly more with a more divisive basic political situation and with Islamic teachings*****.
*Which can't be called Mameluk, tough : while Saqaliba certainly played the role, they weren't a warrior class as you could see in Egypt or India. Furthermore you have a huge part of these forces being comprised of free Berber mercenaries, but as well (especially true in the latter times of the Caliphate) Christian mercenaries from the North.
** I can be wrong and reading too much into this, but I'm going to humor my instinct, wrong as it may be.
*** In no small parts thanks to a really developed bureaucracy. Having political problems that their neighbors didn't have doesn't mean they weren't able to pull something more advanced in the same time in the same field, of course.
**** You'd notice, there as well, the immediate identification of a state as Islamic, in order to illustrate the first and main point about language.
***** One can't stress enough on how Islam is providing not only a moral teaching, but as well legal, fiscal, societal and fiscal ones, more than other main Abrahamic religions and in this case Christianity.
In attempting to fix this, the state may or may not choose a common language in efforts of making the task easier, with Arabic increasingly relegated to liturgical status. Or something.
You can't just separate religious and political features on Arabo-Islamic world : both are extremely tightly unified, the Arab state being an Islamic state (and at the point al-Andalus was a thing, the reverse was essentially true as well).
