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 usually read resumes, but I don't really care about them. Candidates usually send "cover letters" (this is all email, so, really we're just talking about the 2 grafs that precede the resume in the resume email) and a cover letter is enough to get me on the phone. If I like you on the phone, I will never bother looking at your resume.
So going from "I don't read resumes" and "I don't care about your resume" to "I usually read resumes" seems a bit self-contradicting. Regardless, I agree with your point of view. When we were interviewing, we got a bunch of impressive resumes but in reality...
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.
It can be difficult for people in a software management role to come to grips with the fact that they aren't the smartest or most qualified. Its a huge blow to their ego and sense of self worth. They dislike the idea of meritocracy because they are not in their jobs based on merit. One way they act out on these feelings is this 'resumes don't matter' attitude. You see lots of passive aggressive behavior like random destruction of resumes, rejecting candidates over the hiring team's objections, etc.
I have walked into so many interviews where it is clear that the interviewer(s) haven't had even the briefest glance of my resume, that I have to answer "yes".
It is strange how sloppy the interview process can work. I have more than one time been asked to attend an interview with maybe 10-15 minutes advance notice, because someone else did not have time, did not prioritize it, was sick etc. A quick scan through the resume is all that I could do to prepare then.
In these cases, for phone interviews, it is best to reschedule. This happens enough that it is not accidental. It's likely a trick to get the interviewer to give a softball interview.