@8ahwe
This is a tricky one, isn't it? I've got a friend who I have actually known for about 25 years. An old work colleague. We have such a lot in common. She's only about five years younger than me, and we get on really well. We meet for coffee or lunch every two months.
But it's always, always, always me that contacts her to see if she wants to meet up for a coffee/lunch She never, ever, ever messages me first. If I waited for her to message me first, I sometimes wonder if I'd ever see her again. But she does always turn up, and we do have a good laugh for the two or three hours we're together ... She always buys nice Christmas presents and birthday presents and never forgets my birthday. I've got her on Facebook, and she likes and makes nice comments on every single picture or post I put on there.
She's quite busy. She's got four children, aged 12 to 24, (two still at home,) and a job - 28 hours a week she does over 3-4 days... But I'm quite a busy person as well. I mean, I've got children - left home now but I still see them and am a part of their lives... And I have a part time job, (21 hours,) and I've got extended family that I visit and a couple of other friends. and a few hobbies and a husband...
I don't think it would hurt her to contact me first occasionally. I do get a little bit annoyed and a little bit rattled by it - because I do really feel if I didn't contact her again, I'd never see her again.
Also, like your friend, she sometimes takes 2 days to message me back which annoys me a bit. It takes like, 30 seconds to send a message back! Feels like I am left hanging. On some occasions, I wait 3-4 days, and she still hasn't messaged, and I message again to see if she got my message! She messages then, (within about 3 hours,) and doesn't even apologise for not responding for 3-4 days! Just says 'oh yeah, let's meet at The Red Lion at midday on Friday 27th... 
I do sometimes think about not ever messaging her again after the next time we meet, and wait for HER to message me... but I'm afraid she won't. And I don't like the idea of never seeing her again either. So I'm just going to have to take this as a rather annoying idiosyncrasy of hers at the end of the day. I do think she values our friendship, but not as much as I do.
@8ahwe , it really depends how much you want this person in your life. If the fact they never message you first, and it's always you, is the worst thing about them. Then just suck it up and accept it. I have done...