I don't think that's entirely fair.
I think recognizing when maybe your SO needs your support even if it means cutting into video game time is important.
It sucks when you have to wait around for someone to respond to you...I don't think that means this person is making it sound like she's expecting her boyfriend to be at her beck and call.
If someone I was dating mentioned being depressed and needing to talk to me, I wouldn't be like hey sorry, gotta go play my video games for hours and not really ask what's wrong... I'd probably call them and ask them what was wrong. If I was out with friends I'd call and say that I'm out right now but will call soon. If I was in the middle of something I'd say I'll call as soon as I can. That's called being considerate and caring. If the person I was dating said they just wanted to talk and hear my voice, I'd still tell them I'd call as soon as I can. I'd probably say "hey, I'd love to talk, but I'm in the middle of something, I'll call you back when I'm done, in an hour or two!"
She doesn't sound insecure, she sounds upset that her boyfriend isn't being considerate and is making her wait around to get a response, and even then the response is sub-par.