Hi Leah. I'm an adult student who had to take a few years off. A couple of years into lessons I started looking at what I was doing, how, and why and also talked some things through with my teacher.
The first thing I'd like to ask is what your goals are? From what you have written, you want to do the exams. Are they the goal, or are they a means to an end, or part of a larger goal? If you know what your goals are, have you transmitted those goals to your teacher? Do you know what his goals are for you?
You have referred to "some certified board teacher" - I'm not sure what that is. I know that some music teachers are certified by the RCM. I haven't looked into it, but I think these teacher would be examined for understanding the instrument, technique, performance, and also how to teach. You are comparing your teacher to these, finding them lesser. Yet any of those certified teachers might be primarily performers, and might have any number of backgrounds including Juilliard, Curtis, or elsewhere.
Ok, so your teacher has that background, and is a performer. He doesn't write down the pointers. For the things that you do remember after lessons, are they things that help you grow, and suitable to your level and your goals? If you are a true beginner and if you have never studied another instrument, are you getting the basic skills systematically. Or are you just tackling pieces, and being given pointers as to how to make this or that better? are you learning skills such as sight reading, technique? Of course some of this goes back to goals. If you start out without defined goals (like many of us do) and you simply look for the "best" (most credentialed/ inspiring) teacher this can go any which way.
Fwiw, we weren't given written down pointers either. We went home, went through the lesson, wrote our own note and that seemed good enough.
Re: theory. If you are a beginner, you would be studying preliminary rudiments. I think that exam comes at about the gr. 4 level. I've talked to a number of teachers who say they don't really have time to also teach theory. I expressed an interest late in theory. I studied most of it on my own and it is quite possible to do so, and questions that came up were dealt with in part of some of the lessons. I've written all three levels of rudiments.
The thing about theory is that it often becomes a dead dry thing, divorced from music. When you study the formal theory, try to get the reality of it too. If studying intervals, listen to their flavour, find them in music you are playing or hear, experiment with them. Often, too, a musician who is a teacher will teach you theory in a more real way. He may point out a cadence or an effect. These things are more real and truly worthwhile - this is the part we don't get in books. What the rudiment do is to give us the language that musicians need.
Re: familiarity with exams
Our teacher also came from another country. He did have the syllabus and knew enough. However eventually I bought my own copy of the syllabus because it did seem better to be on top of it myself.