For me the "why" of the situation, whether it was a teacher's motivation to build a resume, or the administration's desire to have a certain work load distribution among SE staff, or misaligned stars is of considerably less importance than the net result of having an SE student inappropriately taught for a full year due to a teacher's lack of pertinent training, with no reliable indication that was imminently to be changed.
OP had been communicating regularly with administration and teacher, told to "work with...be patient" with teacher, and apparently was doing so for a year with no measurable progress in the *teacher's* methods or applicable training.
I don't think it unreasonable to question a teacher's ability/willingness to manage her assignment appropriately when there is already a year of less than appropriate effort as history. And I would not particularly care about a teacher's personal or professional motivations where they were observably insufficient to produce a desirable outcome for the child student. OP seems to have been upset more, but not solely by information she was told which may or may not be accurate regarding "why" the teacher was assigned and continuing to teach OP's child - but that information presumably would not have bothered OP at all if the teacher had been actively pursuing developing her own training and ability to teach OP's child or purple students generally.
While I think it would have been more on point if OP had said to the teacher allowing a year for the teacher to obtain appropriate training was all that OP was willing to allow, and the lack of improvement in teacher's skill set meant OP would require a change - OP inferred as much by stating 1. her belief that the teacher was making resume building a priority, and 2. Teacher was not trained appropriately to teach purple students. The important point is the second one, and it appears to have been demonstrably true, with need for training acknowledged by the teacher. The first point, or motivation, is not the reason OP objected to teacher - and may or may not be accurate. Doesn't change the actual issue regardless of motivation.
It is unfortunate that a year of communication with various points did not improve the situation, but an emotional communication to the teacher which led to the teacher's emotional reaction did. As the situation is now better addressed for the student's needs, I still don't see the need for, or appropriateness of offering the teacher an apology for an emotional exchange.
Unless the apology consisted of "I deeply regret that your inappropriate use of techniques with my child, and your insufficient training regarding effective purple techniques led me to become so frustrated after a year that I mentioned a professional aspiration which you may or may not have."