Sneak attack reads
The attack must use a finesse or a ranged weapon...
The ranged weapon property reads
A weapon that can be used to make a ranged Attack has a range in parentheses after the Ammunition or thrown property...
Wild Surge effect #4 on the Barbarian's Path of Wild Magic reads
Magic infuses one weapon of your choice that you are holding. Until your rage ends, the weapon's damage type changes to force, and it gains the light and thrown properties...
As I read this, the barbarian enters rage, and the wildsurge makes his greataxe gain the light and thrown properties. The thrown property qualifies it as a ranged weapon, which means he now has options for sneak attack. He can reckless attack someone with it in melee (Sneak attack's qualifier does not say the attack must be a ranged attack, only that the weapon must be ranged), generating advantage and instantly enabling sneak dice.
He can also now hurl his great axe up to 20 feet at another target engaged with an ally, and also sneak attack them.
Is there an obscure exclusionary rule I'm missing somewhere that prevents this from working RAW, or does this work unless a DM axes it (forgive me, Father, for I have punned) with rule zero?