tl;dr- There are companies that already do a version of this idea. They serve as a method of booking and take a commission on leads and/or repairs. They also change the customer relationship dynamic such that many companies would be wary.
At least one has sent me emails about their service. I found another when I got a Google alert for "iPhone repair (city)." It appeared to operate much as you describe. The customer schedules an appointment or submits info and then the actual shop or repair rep contacts the customer.
My biggest worry was that it adds a layer between customers and my company. In no way did I want my customers to see this booking agency as their repair company. Suppose this booking company was successful and engaged every shop in town. Well, customers would then start to see that booking company as the place to go. The booking company would get the google rankings. etc, etc.
The next worry was that there are so so many variables in this type of work. Most repairs fit within a neat time frame, but I don't want a customer to say, "ABC booking told me it would just be an hour." Well, ABC doesn't know about X, Y, and Z that can cause delays. They didn't know your frame is also warped, or that your battery is bad, etc.
Some repairs have very low margins, some have decent margins. After factoring in costs and such, some repairs basically break even and serve as a way for us to get a foothold in the customer's brain in case they need future repairs or can send us their family or friends. I wouldn't be interested in paying a lot for a lead except maybe for that initial lead. As I said above, the huge worry is that the customer is then YOUR customer and not necessarily mine. Sure, they paid me, but in the future they're going to search for YOU. And, then you might send them to my competitor. No thanks, there.
To get back around to what caused your idea- you're absolutely right that most shops out there have poor websites. My own work's site was terrible before I took it over. I was a little embarassed when I found out how much the owner paid for it. I've brought it into this decade, made it easy to contact us for general info, to get a quote, and even to buy a repair on the site. On that note, customers in this business typically are hesitant to prepay. I don't blame them. You can buy a repair on our site (and the ecommerce platform makes that sleek and easy, with secure payment processing), but it's mostly there as a showroom type deal to show what's available at what price and to give us a professional look.
I think your type of service may be attractive to lone operators, people working out of their house, etc. It may be great for those who just want to focus on their repairs but know nothing about how to setup or purchase a quality website, or marketing.
The trouble is that most of those operators are going to be eaten for lunch by others in their market. The larger fish that survive for more than a few months will probably be savvy enough to see the downsides of your idea.
Those companies may need help with websites, marketing, SEO, etc. But, I don't know if you'd have a lot of luck being their front man.