No, replies Federico Corriente, Dictionary of Arabic and Allied Loanwords: Spanish, Portuguese, Catalan, Galician and Kindred Dialects (2008), it were the Berbers what was the rearguard trumpets, brown-eyed cyclops, antiphons:
Our impression is that Noll has underestimated the cogency of the hypothesis .... purporting that article agglutination has to do with the well-known fact that the majority of the Muslim invaders were superficially Arabicized Berbers who, lacking an article in their native language and being therefore scarcely able to master the rules of its usage, attached it permanently to the [Arabic] loanwords acquired by [Berber], as well as to every substantive in the [Arabic] they learned, spreading this usage in the areas invaded by their troops, the Iberian Peninsula and wide expanses of Western Africa. This adstratic hypothesis is rejected by Noll, pleading an allegedly rapid decay of [Berber] in Al-Andalus, the unaccountable difference of results in the cases of [Catalan] and Southern Italy, where article agglutination is much less frequent, in spite of no lesser contacts with Berbers, and finally the situation of [Moroccan Arabic], where the [Berber] substratum is strongest and, nevertheless, there would be no such abusive use of the article. However, when these arguments and counter-arguments are checked and counter-checked, a different picture emerges, in our view more clearly favourable to that hypothesis.
To begin with, that alleged rapid decay of [Berber] appears to be a myth, originated perhaps in the dislike of many an Arab or Western scholar for an abstruse language which most of them did not know at all, and furthered by the historical fact that many [first-wave Berber invaders] tried at all costs to pass themselves off as Arabs and had no qualms about forging the matching lineages. But this was the attitude of those who, having a chance of socio-economic advancement, dwelt in the cities and played a role in political life. We do not for sure know what happened in the important rural communities of certain parts of the country, nor to what extent they managed to hide from each other a still strong competence in their native language in order to disguise their ethnic background. It stands to reason that, even if they had managed to forget [Berber], they could not help speaking Berberized [Arabic]. No surprise then that [Berber] loanwords in [Andalusi] be relatively few, and even fewer those having entered [Romance], but it is equally well-known that they are also relatively scarce in [North African Arabic] dialects, because [Berber] was and for most purposes still is a discredited language of peasants and highlanders which it is preferable to pretend not to know, even when it is one's mother tongue. However, it can be asserted on mere grounds of population statistics that the majority of the Hispanic people who became Andalusi had to learn [Arabic] from bilingual Berbers...; on the other hand, we have also shown that the situation in [Catalan] is not so different, nor so inexplicable. Finally, it is not completely true that the definite article be used in [Moroccan Arabic] with a distribution strictly governed by the category of determination: the mere existence, especially in non-[Bedouin] dialects, characteristically more influenced by the [Berber] substratum, of an indefinite article of the shape waḥd+ɘl means that substantives not preceded by [ɘl+] are statistically few...
So, for the headline example, the RAE entry for alcalde she say:
Del ár. hisp. alqáḍi, y este del ár. clás. qāḍī, juez