Your question is basically, “Why do tutors use tutoring agencies when they can just advertise themselves and not give away 20-30% of their earnings to these pesky tutoring agencies?”
Firstly, just to clarify: If I advertise by myself without an agency, let’s say I charge £20 per hour to the student. The student pays me £20 directly. Then when I use an agency, I still get £20, but it is the parents who pay the agency an extra £5. That’s usually how it works. So as a tutor, I get the same amount with or without an agency.
Now let me address the question from the perspective of a full-time self-employed tutor, then the agency, then the student/parents. The answer is: because using a tutoring agency is easier and less time-consuming than advertising yourself. It gives you an immediate way to apply to (tens of) thousands of students - in your local area - or otherwise (for example if you want to tutor non-local students online).
You can instead/also advertise in local shops and local schools, facebook etc. Which I also do. But I have to admit that this independent advertising takes a lot more time and effort compared to applying via tutoring agencies. For example, I spend 2 whole days per year driving round my area, putting my flyer up in local newsagents/supermarkets/other shops. This is not an insignificant amount of time. Also, if you want to put your ad in a shop, most of them charge £1+ per week, so this has to be factored into the equation…
So in summary, it is the instant access to thousands of students you are targeting to tutor, and the saving of time and the ease of getting new students that you are paying the tutoring agency for.
As for the tutoring agency’s business model, it’s pretty obvious to me that it makes sense from a “maximising profit” perspective. The agency gets a bigger profit than a one-off payment from the tutor/student in order to get their contact details. Also, sometimes the student (or tutor) decide to discontinue with one another after the first lesson or two due to not being on the same wavelength/other reasons. So what should the one-off payment be? A large one-off amount doesn’t make sense for the tutor/student and a small/medium amount doesn’t make sense for the agency compared to the 20-30% for each lesson.
Finally, from the perspective of the students/ students parents: there are tens, possibly hundreds (especially due to the Covid lockdown began in March 2020) of tutors on the agency website. The students and their parents can view the profiles of all these tutors on the tutoring agency website, and decide which one is most suitable for them. That is the main thing the student/their parents are paying the agency for. Therefore, if the student/ their parents don’t care too much what type of tutor they get, or if the parents are frugal with their money, then I think it makes more sense for them to avoid tutoring agencies and to look out for an add in the window of the local newsagents, or on facebook. Else if they do care, or if the student is very particular in what they need (especially if they have specific learning difficulties or tutor quality requirements), then using an agency is probably preferable to not.
I’m not sure what you mean by, “I can accomplish the same thing by using Zoom”. Accomplish what?