Should men always initiate?

I'm in my mid-20s and pretty attractive with the occasional quirks. I don't think I have a hard time meeting men -- I've been on Match and even met someone on a plane ride home once. I think dating is fun as long as you have the right mindset. And I definitely have my fair share of funny stories, but nothing ever comes out of these two-to-three-date men. I'd say it's definitely a mix of both myself and them -- I either don't like them enough or they're just not as into me. I haven't gotten particularly heartbroken over any of them, except for one. We had a fun fling that turned serious (over a year-and-a-half), but it never became an actual relationship. He had never done that before and was scared, plus he was still trying to get his life together (job, living, etc.). Eventually we amicably cut ties. It's been a few months and I recently reached out to him because he had been on my mind. We've hung out twice, including one sleepover, and he's told me he's missed me, but while he was drunk, which is the only time he's ever really open with his feelings. After that I mentioned to him that I'd like to see him again, and he agreed that he'd like to see me too. I have no expectations whatsoever at this point since the relationship never turned serious even though we had serious feelings for each other.

OK, so here's my question for you and the readers. Most of my girlfriends advise me that the guy -- no matter what -- should always text first, call first, reach out first. Always. I'm all for playing hard to get. But I feel like by completely not making any communication effort is playing a game. I'm not into games. But I'm obviously doing something wrong here since I'm still single. Did my reaching out to him first (both times), mean he's just not that into me? Should I never do this with future dates? I was trying to be open and show my interest, but does this rule always apply? If the guy doesn't message you a day or two after you hang out, does that mean he's just not that into you? Do men always go after what they want, no matter how foolish they may look? Or how long it's been? Or how shy and scared they may be? I don't believe all men are the same. But this "rule" says differently. To play devil's advocate, wouldn’t this also apply to women? If I really like a guy, why not shoot him a message and ask how his day was or if he wanted to hang out if I was in his area? Is that a turn-off?

– Gray Area, Massachusetts