Rarely enough for me, maybe a once a year if that (approx 1000- 1500 miles/yr).Out of interest, are you getting punctures with those Gatorskins?
Prior to the ebike I've used them for about 5 years on a standard touring bike with few problems.
The punctures I've had are nearly all down to hedging detritus sprayed across the roads - hawthorn is the killer. Second only in hardness to diamond, I reckon. Ask any cow or sheep!
I rarely ride in town, so I can't say how they work on glass/metal crud, but the country roads round here have some pretty rough patching and potholes which you can't always dodge, and they seem to work OK there.
On the ebike (bought Apr/May) I've had one puncture - a 'snakebite' first time out, and that was all my own work for not pumping them up hard enough (keeping on-topic!), and then heading off down an offroad section near us. Since then and going up to 100psi (from 80psi) I've had no problems.