I said that because it is utterly pointless to get technical with grammar points on here as everything you have said in this post could have been lifted directly from the internet. In fact when I get to the office later I will check that very thing. My assessment of what it is is correct. Not tricky at all. You asked a specific question, I gave a specific answer and as usual you are trying to convolute that. As you have been wrong countless times with your own use of English and attributed it to descriptive grammar, it is hypocritical of you to then trot out grammar rules for use of English to score points. Either grammar has rules that should be followed or it can be used willy nilly to mean anything you like. It cannot be both.

I have been teaching for 11 years and not once have I used the phrase " accusative cum infinitive" with any of my students, including up to proficiency level as the the use of it should have been acquired by A2 level at which level they do not need to know terminology, just how and why it is used. I would teach such points under the umbrella of subject and object pronouns, which is when that type of phrase would be useful.

No, a native speaker without qualifications is not qualified to teach grammar. But then neither is a native speaker with an unrelated degree. If they both have a valid classroom based TEFL certification, I see no reason why they can't teach conversation classes in countries where there is more of a demand than can be met by qualified teachers. I do not like it, but unfortunately nothing can be done about it. Yes, grammar points may come up at the lower levels, but they shouldn't be teaching those classes, they should be teaching B2+ speakers looking to refine their already acquired ability in discussion classes. The real ESL teaching is done well before that, hopefully by real qualified teachers.

Non-native speakers run a distant last to qualified native teachers, native speakers with unrelated degrees and a TEFL and degreeless native speakers with a TEFL... You may well have a degree in a related subject but that doesn't change the fact you will never have the natural fluency of the native speaker, who would never make mistakes with phrasal verbs or the like as the use of them is completely natural to them. They also have much better pronunciation. These things are more important to an ESL student than thier teacher knowing the term "accusative cum infinitive" especially so when the way in which you've explained it here would confuse the hell out of the students at the level this point would be taught! I've taught this successfully in the past to all ages at A1-2 level using stick figures and colour coded tables on the board, after which they make a comic strip to practice all the variations! Absolutely no need whatsoever to talk about embedded agents!