Short answer: of course you can! Anything can be a bayonets, the differentiation comes from the entrepreneur in securing that market. You have, in a microlevel, proved that there is a need and is just a matter of how big that market can be.
What I would do, if in your shoes is keep the idea simple. Target only one industry or type of phone call in which there is in fact a benefit of having a third party listening to a call to then offer suggestions. The phone calls that you guys made could be the type of call to focus on! But it has to be one where the feedback is the only service to the customer.
Don't get too hung up at first about monetizing, once you find the right type of call you can then focus on how to monetize. You can charge businesses within that industry or charge customers directly.
If you charge customers directly maybe have a prepayment processed similar to clarity for example.

If you need more feedback give me or another a call
Best of luck!

This is a great coaching process; I had help hundreds of people to sell to new customers. Usually I was seating on their side participating as a support person for them, taking notes and watching how the conversation went. I always wanted to tape the conversation, but this was not done because requires the authorization of the customer, that do not know my role and less that my mentee was under training. I wonder if you make it as an open sales training session to disclose to the potential customer in the same way that a call center tell you about the recording of the conversation and you help your customers thru a website that create the sales feedback training process.

For me sound great a idea, that needs some work to find the best way to go.

This would be an interesting business indeed. You need to be careful with laws in varying states however. The reason why you are always warned that there may be someone recording or monitoring your call for "training purposes" is most states require both parties know that a recording device is in use. When starting up a three way call you need to disclose that there is a third party that is listening in on the call.

However using the same line for "training purposes" is actually a great way to do it. Something else you can do is use Google voice since it will record your phone calls for you. The catch is they have to call you for Google to record the call (again this is for legal purposes) but all you will have to do is press 4 and they will record the whole thing.

In this kind of a business you can easily have your client call you and then you call the prospect so you can record the call. Just make sure that the client alerts the prospect that the call "may be monitored for training purposes" which is completely true in your case.

As far as marketing goes I think a great place to start would be to find small business owners who use cold calling as a way of getting new clients. Then I would create a package where you monitor X number of calls and then go to their office or a skype call or Google Hangout or something and train them on how to do better. I would also suggest having written material they can keep as well. You have a fantastic idea!

Let me know if you need any help in marketing or setting up the business.

In business, especially sales the idea of cold calling usually makes most of us sweat profusely, get clammy palms and occasionally hyperventilate. You've demonstrated a proven way to help someone through this anxiety with giving immediate feedback as the individual (your client) makes the call. This could be considered a type of mentorship, business coaching or as you noted, accountability partner. My question for you is, what did you do to help this individual? Can you identify a specific method, something new that others need to know now? There is your business model and your method and your product.

Giving someone the capability to have confidence to make a cold call, take a meeting, do the things they need to do promote their business is definitely a skill. How you convey the message, how you structure your business and what other wonderful strengths and gifts you possess are something you should definitely explore more. Do some research on the market, is there a need for this type of service, what space do you want to work in, what kind of package could you offer? Practice coaching calls, webinars, group coaching, role playing these types of scenarios to help with the calling process? I encourage you to consider having a conversation with a friend, a coach, a business mentor and talk about your ideas. Feedback on what you're thinking can help you more clearly define what type of business you want and how you can continue to use your skills to serve others.