Surely you're joking--Notch gave away the POC for free, and started charging for the beta somewhere before 2011 (when I started playing). Since then, new features and bug fixes have been essentially non-stop.
I picked it up after a long hiatus and found new terrain types, new monsters, new neutral creatures, an enchanting system, a brewing system, an entirely new dimension ("The End"), and several types of abandoned structures.
Minecraft's price has increased slowly as the game got more and more stuff, and the pace of improvements and additions has increased significantly over time.
Minecraft today is $27 (and totally worth every penny and then some), but this is very much not what Minecraft cost 2 years ago. The alpha version was €9.95 and the beta was €14.95.
So no this is not at all what happened with Minecraft.
Yeah, this is the way to do it, and I hope it starts to get more common on Steam and other platforms.
Charging full price for alpha versions of your game, then putting it on sale once it's done, is a slap in the face to your biggest fans - and the people who will most likely help you succeed via word of mouth.
It seemed like this happened with Minecraft, it was just popular enough that the POC was good enough.