I think it only charges when the camera is off and of course for transfers it will be on.
Yes, this is true, at least for my X-T2 and X-T20.
Is it true that numerous short re-charges are bad? I'm not so bothered about the camera, but I wonder about the wife's car.
Typical car batteries are lead-acid, they are not affected by memory effect. However, if your trips are too short, the alternator may not be able to recharge the battery to compensate for the energy lost to starting the car, eventually you would need to jump the car or charge the battery manually. They usually last only 3-5 years anyway.
Camera batteries, Li-ion, are not supposed to be affected by memory effect. However, they best stay in 40-80% range, frequent short cycles of charge-discharge in the 90-100% range would kill the battery quicker. Recent laptops have a battery-conservation mode for users who keep the laptop plugged most of the time. The laptop simply keeps the battery at 60%, instead of topping it at 100%.
My concern came from using the X-T2 with USB power bank via the microUSB cable, as the 50-140/2.8 drains battery 3 times faster than other lenses. While camera is on, there is no charging, but the battery also discharge slowly as the USB power isn't enough (might not be the case with USB-C). When camera is off, the battery is charged from the USB bank. Thus, during a whole 3-hour event, I turned the camera on/off rapidly, subjecting the battery to rapid discharge/charge cycles. Together with the heat from using the camera, the battery got swollen very quickly.
Plugging the camera into the computer is not as severe as the above scenario, but users should keep the above in mind and not subject the camera to multiple on/off cycles while plugged, especially when the battery is already full (not sure if the camera is smart enough to not charge the battery if it's at least 90% full).
Besides, the card slot is much less likely to get damaged than the tiny delicate USB-C port. Using a card reader is a better long-term option. Worst case, if the cable goes wrong or the pins of the USB-C port get shorted, the camera main board will need to be replaced.
For charging, looks like Wasabi has USB charger for NP-W235, those are super convenient on the road (sharing USB power banks with other devices):
https://www.wasabipower.com/collections/fujifilm-np-w235