I had this conversation last night with a girl who had just gotten out of a six month relationship she had started with a tinder date. And one question she had, which I've heard a lot from girls, is "Why after we match does the guy not send a message?"
It's totally right that it is the guy's job essentially to send the first message. And guys WILL do this, but they basically do it in a queue. Each girl gets messaged in turn. So if you happen to be a slightly worse potential fit, further away than walking distance, or simply match him later in the day, he'll start messaging you after he finishes with the girls ahead of you in line.
The good thing about this is that you know that while he's messaging you, when he's actually being serious about meeting up for drinks, he's not messaging a bunch of other Tinder dates at the same time. We do this because IT IS EASY TO GET DATES ON TINDER. You already know they find you attractive (probably based entirely on your height). So if you message two girls, you will end up having to tell one of them "Sorry, I can't meet tonight", which seems like a waste of good flirting.
This is one of those things that probably changes as guys get older. I've seen younger dudes message eight girls the same horrible pick up line and just pick the first one that gets back to them with a number. Guys in their thirties who've used Tinder more than a month generally swipe left as much as you do, and are generally busier and more likely to only message you when they actually want to meet up.