Did the Aztecs have ritual specialists?

Yes, the Mexica had plenty of ritual specialists, from midwives to New Fire Ceremony priests, from astronomers to deity impersonators to sweat bath practitioners... Perhaps the most ‘obvious’ one to choose is the Aztec soothsayer or tonalpouhque in Nahuatl, since his services were called on constantly in daily life. Here’s a brief intro, from the classic book by Jacques Soustelle Daily Life of the Aztecs on the Eve of the Spanish Conquest (1970, p. 56):-
’They were educated in the monastery schools, for it was there that a knowledge of the characters used in the divinatory calendar was taught; and indeed this knowledge formed an integral part of the higher education... Soothsayers did not become members of a temple when they were qualified; they set up on their own account. Neither business nor income can have lacked, for every family necessarily went to a soothsayer whenever a child was born; furthermore, there was no important occasion in life, marriage, leaving for a journey or a military expedition, etc., whose date was not fixed by the soothsayers, either at the request of private persons or of officials. Each of these consultations was paid for by a meal [and] by presents, “several cloaks, some turkeys and a load of victuals”’.

One of the double-page sequences in the Codex Cospi (facsimile), showing a ‘trecena’ calendrical sequence