Is "I don’t read resumes", "I don’t care about your resume", very common? It seems a bit cold hearted and unfriendly. It says to me "I don't care enough to read a bit about you, and your achievements". But that's just me.
I do not care about resumes.
If I get on the phone with you because of your cover letter and haven't read your resume, I will probably never read your resume.
Regardless of whether I ever look at your resume, it's very unlikely to have any impact on our hiring process. We use practical tests and, to a much lesser extent, 1:1 interviews on concepts; the practical tests have been so valuable that we're moving towards doing more of them and less subjective interviews.
Your resume fits almost nowhere in this process.