Thats a nice bike, should make a great build
As for the motor.. 680 watts, Thats an odd number for the company to come up with. Usualy companies that do that are giving you the max wattage, based on what the controller will allow, and rounded up. a 36v (42peak) LifePo4 battery with a 16 amp controller would have a peak wattage of 672 watts.
Most reputable companies rate there motor based on its constant load, not its peak, but there is no actual guideline. very likely that makes it a 500 watt motor by the conventional measurments we use here. The other hint about it being a 500 watt motor is the size of the controller they offer. its tiny. 6 fetts at best, which makes it appropriate for a 500 watt motor at 36 volts.
So what does all that mean? Maybe good news for you, and heres why:
A reputable company wouldn't go through the expense of buying the millions of dollars worth of manufacturing facilities and equipment, then make grandious claims that are only true by a technicality.
That would be like Ford claiming all it's current and past cars actualy run on a renewable resource and are powered by the sun. Technicly true, oil is renewable if we give the earth afew hundred million years, and the original energy in the oil came from sunlight photosynthisized by the plants before they died but thats a stretch so long Ford wouldn't dare make it.
So that means Pheonix is likely a reseller, buying there motors from someplace else, then developing them into a kit. There aren't that many manufacturers of Brushless motors, as the equipment for them is extreamly expensive right now, and the technology is still so new. So the motor is likely from one of the major manufacturers and therefore likely to exhibit similar characteristics.
In otherwords, yeah, the motor will very likely take 72 volts without issue.
The only problem you might run into isif the motor was speed Wound for 36 volts. if thats the case, it will be an Amp hog at 72 volts
Whats it's top speed right now?
As for controllers, i would go with a 20 Amp 72V Infirion. that would give you 2 horsepower peak, enough to do a burnout with your weight off the bike.