I have never met an ESL teacher who has a degree in a language related subject

Understanding the world based solely on personal experiences begins at birth, but ends for most individuals when they learn to read.
As an EFL teacher, TESOL certified by a university, with an undergraduate degree in English, I am able to Teach integrated English language skills to Students of Other Languages. The FTs you "have met" at the expat bars and massage parlors routinely cannot.

Academics on this forum may be fully capable of explaining the mechanics as to WHY we never
say something such as 'I very like', but how important are those mechanics to an ESL student?

I would very much like to tell you the answer: non-native language students that desire to speak English correctly must acquire grammatical information concerning the use of adverbs with modal and non-modal verbs. Why? Even limited vocabularies create innumerable combinations of words; the "mechanics" are necessary to understand what combinations of sentence components are used to comprise native language speech and why.

You likely have the ability to speak your native language somewhat correctly. Dancing FTs in China model their own native language somewhat correctly while Chinese teachers provide students the necessary grammatical formations. An ESL/EFL certified teacher has the ability to provide that grammatical information.

This Dancing FT/ traditional Chinese PPT English lecturer model of education doesn't work very well. Chinese educational authorities recognize this and are changing regulations to limit Z visas to qualified FTs and provide improved language education skills to their own teachers.

The days of Dancing FTs are numbered.