It can run a Pi 3, but you'll get lightning bolts. From what I've seen and experienced, the voltage drops/lightning bolts can mostly be eliminated by using two batteries. (which carries the possibility that they'll charge unevenly) I suspect using a really good battery (like Helder's MegaBat viewtopic.php?t=6720) might do the trick, but I'm not sure on that. I think the resistance on the mosfet switch might be just a little higher than it should be also, and I've been tweaking the design to see if I can change that. I should have a new batch of the "Power Assist Basic" boards arriving next week, and if they work like I'm hoping, I'll probably order the production batches for all of the boards using the new design.
For now I can't really recommend it for use with a 3, unfortunately, especially if it's paired with a power-hungry composite or HDMI screen. The power drain of an SPI screen is small enough that you'd probably be okay pairing it with a 3, but you're still likely to see the lightning bolts occasionally. I'm hoping to be able to change this recommendation soon.
I've been working on updates to quite a few of the other boards also, and I've also got some new items in the testing stage that I hope to release soon, so I hope you'll stay tuned.