I very much get the impression that the rink/club is terrified of offending anyone (or turning away anyone) for any reason.
Then maybe they won't turn you away if you ask to be a professional coach.
But give it a little time, so you get experience teaching as a volunteer, which is like being an apprentice coach, and so the figure skating director gets to know you. If you are lucky, they will give you pointers on teaching. Make a point of getting to know the director, and show her that you are serious about helping, and are doing your best. Ask for her advice on things, to show your interest.
I did ask a director at Fort Dupont whether I could coach at that rink, and she said no. But I'm a much less advanced skater than you. There were other volunteers in the program who wanted to coach who were better skaters and could do a better job than me.
I've noticed that some new coaches find the most difficult issue to be dealing with students and their parents who are less devoted to skating than they are. E.g., most of them will not join ANY club, unless they have to, to take lessons. And many, maybe most, will miss a lot of sessions, and/or come late.
Another thing some coaches have trouble with - they are so good at athletics, that don't really understand what it is they are doing to make stuff happen. You may have learned how to move efficiently when you were very young, and are unconsciously unaware of what how you move. Some of what follows is my personal teaching style, but you may need to help students who have trouble think about things like:
Many beginners, encountering the balance difficulty, freeze ALL their muscles (activate stabilizing muscles all over their bodies) with fear so hard they feel like they don't have the strength to move at all. They unconsciously think freezing will stop them from falling. They need to relax. A lot of people are so afraid of falling that they can't relax. Which is part of why I spent extra time showing them they could fall in any direction without getting hurt. I sometimes start them falling from a sit position on the ice, advance to being on the knees, and then to standing. I have them practice falling forwards, sideways, and back, in appropriate body positions, so none of those create fear. At Fort Dupont, we always started fall practice off ice, perhaps to remove the fear-frozen muscle problem. Make it FUN, e.g., a game. They also need to understand that being too stiff makes them get hurt easily, that a relaxed fall, done right, doesn't hurt at all. If you have any adult students, that may be a lot harder lesson, BTW, one some will never get. (OK, I'm a fall practice nut, and would love to teach just that.)
Think about which muscles you use to do various moves. Which muscles actually move, and which freeze to stabilize other joints, to make the motion smoother, more controlled, and more efficient, and to transfer momentum from one part of the body to another? Kids won't know the names of the muscles, but you can show them where you feel stuff tightening and stuff moving.
Because most of the kids don't learn motions well from words alone, but are pre-programmed to the monkey-see-monkey-do thing from birth, it helps to simultaneously say and show them what you are doing - e.g., point to where the muscles are that are doing things, and then actually show the move, so they can see what you are doing. In a class, the kids will have individual learning styles - some listen, some watch, some do both - try to find ways to teach them all at once. And making it FUN is a great way to teach anyone.
"March and glide": Many simply won't get that - that they do need to push sideways with the pushing foot a little, in order to glide. If all they do is pick up their foot and plant it forwards, they don't glide at all. That will segue naturally later into skating technique, but you need it a bit at the start too. A sideways push and weight shift IS often/usually part of efficient walking and running technique, but most people don't know it. Typical running technique, in shoes with a cushioned heel: land balanced on your heel, roll outwards to midfoot, roll inwards to push against the big toe; the rolls and weight shift typically provide more than half the power, because they make use of a set of core muscles that don't get used much if you just bend and unbend your knees and swing your legs forward and back. Of course it is a lot more subtle in shoes than skates, because straight point feet just slide forward and back without push much on the ice (show them!). But if you point the weight transfer thing from walking and running out to them, it may help them relate skills they have already mastered to what they are learning now.
If someone is pushing and pulling sideways really hard to do swizzles, tell them that if they bend down deep at the start, their downwards motion will naturally push against their knees and push the feet out,
without significant felt effort, and if the straight up for the second part, that will naturally pull in the feet, again without significant felt effort. Later, power pulls can work the same way, though the timing is more complicated. Although it feels effortless, this is an early case where tightening the core muscles a little to stabilize that core seems to help.
I said relaxing is important at first, but eventually, tightening up your core muscles is essential to balance, enhances control, and makes motion and momentum transfers more efficient. Again, you may not be consciously aware of this, and some of your students don't need to be either - but some do. Point out that they do it off-ice without skates even when they stand on one foot. That relates to something they already know how to do. Show them if you relax all over, you collapse in a pile on the ice. Let them try! (A fun move.) Show them that if they tighten their core muscles, they regain their balance.
Think: How do you initiate a turn, in detail? You were probably told to bend and unbend your knees to de-weight the foot or feet on the ice , point your toe, and twist your lower body against your upper body, and maybe you learned to push against the edge. That's a lot to synchronize. The simple act of turning your foot outwards into the turn, so friction against the ice from your forwards motion pulls the rest of you effortlessly into the turn, is so obvious to good athletes that it is rarely pointed out - but it may be the easiest way for a newbie to initiate a turn. I.e., you get moving, turn the foot out (turn at the hip), and that makes you turn. They should learn the other things too, though they take more coordination and maybe strength, but until they do the simple foot turn most real athletes are not consciously aware of, they may be unable to get started. Perhaps they won't need the foot/hip joint turn eventually, but it sure helps at first. I don't spin well, but the pre-turn into the motion this helps initiate spins too - in some ways a spin is a turn without the check, or with a much-delayed check. I'm terrible at checking, but in theory, reversing the foot turn at the hip might help
a little, by again pulling against the ice, this time to stop the turn. In fact, if you leave the foot turned out, you will keep turning, and checking the turn is almost impossible. Again, that may be so basic, you are not be consciously aware of this - try it.
They can't do crossovers smoothly? Show them the crossing leg isn't long enough to touch the ice unless the skating leg bends, and actually gets onto an outside edge. A lot of kids, and adults, try to keep the leg straight, or stay on an inside edge, and then abruptly fall onto the crossing foot. It is hard to make that smooth. Hockey kids learn to do it on inside edges, for stability when being checked, but they lose a efficiency doing it, and it isn't right for figure skating. Again, make it a game.
See the general idea? Think about what you do in detail. Say and Show. Relate new things to old things. Add in silly things that help them learn, like the relaxed falls. Make it a game. Smile. Have fun. Let them have fun. Don't take it too seriously - most of them won't, and you should just accept that. By the end of each class, do some simple routines, and take turns letting some of them lead games like "Simon Says" and "Monkey see, monkey do".
By the way, PSA materials are really great. Read them.