It's impossible that a third party like yours can do that if the channel abides on YouTube terms and agreements; you don't even outline criterias on what channels you can delete
We have team of lawyers, they make strong case for the removal of any particular video and yes there isn't any particular criteria for videos for removal. it depends on case. if that video really creates problem for reputation for client then definitely we delete that video or channel.
I don't think you need lawyers to do that, YouTube is a community-based platform and their criteria are clearly outlined, If you're reputation has been tarnished, you just have to create an appeal by following their guidelines .
There are no rules or requirements coming from YouTube that require you to get a lawyer to prepare your statement; in fact, if you follow these guidelines, your request will be granted within 24 hours.
Your post is deceptive; you make it appear that YouTube is like a court where you need a lot of formality and you need to spend hundreds to thousands of dollars to take down a video or channel.
Report inappropriate videos, channels, and other content on YouTube