I don’t think it is crazy and I have suggested beforehand there needs to be some sort of proof of work on the candidate side to prevent resume spam.

I think your idea is very elegant as everyone has access to the mail system, an actual stamp is pretty cheap, but it is just enough hassle to mail an application that it will filter out some of the spam.

The other suggestion I have had is that candidates need to hand in the resume in person, but I guess you could accept resumes from both mail and in person drop offs.