While this sounds reasonable, and it may be quite true in general, Harvard has the earliest deadline of any grad school I know; certainly it had the earliest deadline by far of any school I applied to. I have good reason to suspect that your chances of being accepted are fairly constant up to the deadline (down to the minute), at which point they drop to zero. I imagine that Harvard GSAS puts the deadline early so that they can have the luxury of this policy.
As someone who just completed this process last year (college freshman, whoo!), it’s not.
They encouraged us to apply early (decision or action, but few schools offer action), and there was really no stigma with the common app. I did early decision and used the common application, and I got into Barnard. 
It really does tell the school it’s your first choice, and since it’s so hard to get in, it increases your chances. And you are going up against the best candidates, but let’s face it, you’re going up against good candidates anyway. Hell, if you’re applying to Harvard early, you are most likely a good candidate…
Best of luck, and good hunting.