Try raking the bottom of the lightning socket with something thin and stiff, such as a thin paperclip (or use the Probe and Pick Set) to remove the lint.
I had a similar issue with my iPhone 5S and the Apple Store guy took it back for 3 minutes and came back and all was well.
One giveaway is that the white plastic collar (for lack of a better term) doesn't sit flush against the phones body when the lightning cable is plugged in. Once that lint was removed, it was flush again and charging was back to normal.
Toothpicks (wooden tools) are safer but may not tough enough.
I used metal tools but was careful not to mess with the metal pins.
Yes, a toothpick (or anything wood) is certainly a lot safer than metal tools. The toothpicks I've used aren't hard enough to break up the cruft at the back wall of the port.
You could use a teflon spudger as an in-between that's stronger than wood but can't do damage like a metal pick would.
That being said, I recently had to clear out the lightning port a second time (on my iPhone 6s) and I used my metal tools without issue. The pins are located on the floor/ceiling of the port chamber so I mostly kept clear of the pins by sticking to the back wall to break up the packed cruft.