A few things here:
Consistent work. I can fill their schedules very quickly. 5 hours a day at $40/hour is not bad.
While I'm a trained teacher, I am in a different profession as a solopreneur these days. Some similarities is that we have people who practice the profession without training in the field, at a less professional level, and others who are highly qualified and competent. It's the same confusing scenario as private one-on-one teaching.
In my profession, I can try to get my own clients (and I do), or I can work for a middleman - called agencies or companies. They're well advertised, very visible, and have an admin structured centered on interacting with interested clients. And so theoretically, there's an advantage to go with them (same reason as yours). These middlemen contact me, and offer me "consistent work" - but I need to lower my fees substantially. Sometimes it's 50% of what I charge. My argument is that not only do I need to work twice as hard to earn the same amount of money; but with "regular, consistent work" keeping me busy, I'd have to say no to my clients who will pay me double. And having built a reputation, I do have those clients. It is the inexperienced novice who does not yet have the reputation and clients who is most likely to be enticed. I suppose the same is true in your case.
The $40/hour - isn't, really. I mean, I've taught, both in the classroom and later one-on-one. A decent teacher often does prep work. That one hour lesson may be 90 hours between the lesson, and the preparation of material and whatever. If the teacher responds to student questions in between, maybe altogether it may be 2 hours all told.
I handle all the parents and payments so they don't have to deal with that.
That of course can be a plus.
Well, let's say I have a recital. My students go up and perform their pieces alongside the students of the other teacher (who I hired). There should be some relative consistency in the teaching results right. Otherwise the parents will be scratching their heads and wonder why one group is significantly less developed than the other, yet everyone is paying the same price.
Here I had a note of caution. On the question of expectations there are many things you didn't list. Such as skills.
About recitals. When my child started lessons, initially the teacher taught at a "place". There were biannual recitals. The teachers were pressured to show "fast progress". I remember the 6 year old who broke down trying to sing Ave Maria. Our teacher went slower, and solidly, by comparison. What I heard, as someone with an ear, is that this "easier" music was being played well, with assurance - the kids appeared well taught. But that might not have been the overall impression.
About recitals again: A child can be rehearsing a small number of pieces - maybe only one or two - and then shine in the recitals, giving the teacher and the school a stellar reputation. Later as a transfer student, the new teacher discovers a bunch of things this child never learned to do. But boy, those 8 pieces learned over 4 years, aren't they amazing!
You are right, however, that credentials - pieces of paper - will not necessarily reflect the quality of the teacher or that the teacher can teach. Those of us who are students are faced with that same problem.
For your situation, do you have teacher right now whom you have retained - who have stayed with you for a couple of years and thus must be satisfied, who could try to pass on to good teachers they may know, that you have open spots?
An afterthought: From time to time we hear from teachers who teach at a "place" and sometimes they don't seem to have freedom to choose. For example, they are assigned students and don't have a say. Or may be told how they should teach, or what. Such a thing might attract learner students, but be a no-go for seasoned professionals.
Just throwing things out there.
Topic: Need Teachers!