You're not recapping his story correctly. Here's what he said:
See the part in bold? That's where he went wrong - she didn't stop texting him until HE started texting HER too much and came across as desperate and needy.
Real talk: in a relationship, there are just some things that girls can get away with that us guys shouldn't do. Women can call, text, message us all day and it not bug us because, to us, it shows interest. However, men can NOT do the same thing to a woman - it comes across as non-masculine, being needy, un-trusting (i.e. always texting her asking what she's doing and/or who she's with) clingy... the list goes on and on.
The problem is, you're trying to relate to a woman as if she thinks like a guy. Real talk, there needs to be a book called "Think Like a Woman, Act Like a Man," 'cause the way you and Dave are reacting to these girls when they allegedly complain about something is all wrong.
In your situation: if a girl is saying stuff like "why didn't you reply back" or asking if you're busy, it means she has high interest. However, your response to these comments shouldn't be to change your behavior and start writing back/contacting her more, but to keep doing the EXACT same thing you've been doing so that her interest remains high. All the girl is saying is that she's frustrated because, when you don't contact her back, she doesn't know what you're doing. And she's DYING to know, but since you're being a challenge and a mystery at times she can't quite figure you out. And you don't WANT a girl to ever figure you out COMPLETELY because once she does, the fun is over for her - she knows everything about you, becomes BORED of you, and starts looking for someone else.
I'm telling you - STAY OFF THE PHONE AS MUCH AS POSSIBLE. Use it to set up date times and places, and that's it. At the very least, do this until the girl is your girlfriend, and even then, keep texting/calls/messaging/etc. to a minimum. Always strive to have more face-to-face time with a chick, and leave the things you'd talk about in a message for your dates so you'll actually have stuff to talk about when you meet up. Hope this helps!