First of all, it sucks this happened to you. The DMCA forms they have you fill out are required per US law as I am sure you know and this is why they require them to be done.
As for the “crowdsourced” DMCA monitoring (which I would argue is not what they are doing as the only one that can actually submit the DMCA notice is the owner of said work), how else would you expect it to be done? Even a company like Google works this way. Google does not proactively decide that a video submitted to youtube is infringing, it has the content uploaded via the copyright holder and then matches the videos against that. I am assuming that you have not uploaded your videos to Udemy for them to match against.
As for the verifying of the identity, even though it has your name, the laws still require this to be done.
While I am sure that Udemy could do more, I would love to hear what you would like to see done that still keeps them legally in the clear and is reasonable to be implemented. I would say that having a < 12hr turn around on a holiday to a DMCA complaint would make most people happy.