But it is close.
Personally, I wouldn't trust it. It will probably be fine, but it is definitely cutting it close, IMO. PSUs are pretty cheap, get a good one and you won't regret it, get a cheap one and you might...
And power supplies aren't really that expensive as it is, just get a good one and not have to worry.
They aren't going to be that much different. In fact, from what I've seen the Zotac 1070 Mini actually consumes about 20w MORE than a founders edition 1070. So probably about 165-170w, and that is before adjusting the power limit.
To me, the cost of a good power supply is cheap compared to the price of that 1070 and the rest of the system. I've seen too many components lost to cheap PSUs to recommend anyone use them, especially with higher end components like a 1070.
Will the current PSU be good enough? Maybe. Would I be willing to risk it to save $55 on buying a good quality PSU? Nope.