Blocks/Entities have very complex and different behaviors so I'm not sure how they would pull that off. JSON is designed to store structures, not to describe logic. For the logic (and many other things) they would probably just expose an API in whatever language they want.
Oh, I grant that a block done through JSON wouldn't be too fancy - that would require code. But if all you want is a block with specified texture, hardness, blast resistance, illuminance, transparency, redstone power, recipe, and drops, that would be quite easy. And that covers an awful lot of blocks out there.
You could also probably specify certain actions, even commands, when it's clicked on or the like. But anything more complex than that would need code. You're not going to create something like the comparator just with JSON, for example.
EDIT: Likewise, we've seen with the add-ons that you can do quite a bit to alter existing mobs with JSON. All we would need is a way to register a new mob, even if it has to use one of the existing models, and specify where and under what conditions it spawns.
The new Advancements clearly aren't finished yet, so there's no point in comparing them to Achievements yet.
I suspect we'll see some, though not all, of the more distinctive Achievements reappear as Advancements. _Diamonds To You!_ was always rather silly, and I don't see much use in the sort of triggers and conditions that would be necessary to make it work.
I'm a little curious as to what new triggers will be necessary to duplicate _When Pigs Fly_. There's currently a trigger for a player killing a mob, but none I can see for mobs dying from non-player causes.
_On a Rail_ would in principle be pretty easy to do, but they haven't laid the groundwork yet.
_Return to Sender_ could be done right now, but is probably waiting on a Nether tab.
Likewise, _The Lie_ is quite doable, as are several other food achievements, but they're likely waiting on a Food tab.
It's always annoyed me that there's no achievement for breeding a mule, but that's really easy to add now.