Do the items drop only in level 70 rifts? Or do they drop everywhere after finishing a level 70 rift?
From the patch notes for Update 2.5:
Note: Primal Ancients will only begin dropping after a character on the player’s account has reached Greater Rift 70 Solo
So you must complete a solo 70 greater rift to unlock the chance for them to drop, but they can drop from any source (regular loot drops/bounty caches/upgrade through the cube, even Kadala!) This must be done for each game type (once for Hardcore, Softcore, and Season/Nonseason of each of those).
I can personally attest to getting a primal from Kadala (gambling with blood shards). Now my Templar has a real nice Freeze of Deflection.
And here is video evidence of the cube creating one as well.