Yeah, they do backlight strobing (LCD) or black frame insertion (OLED), to reduce blurring during smooth eye movements, at the cost of overall screen brightness. I actually think small CRTs would be perfect for VR headsets in this regard, as they are naturally have very short frame persistence.

One likely problem for battery powered headsets is the (I believe) relatively high CRT power draw. Another is probably the fact that they aren't used for anything else anymore, meaning CRT development has stopped a long time ago. There were quite small CRTs in the past for special applications, but probably not as small as is optimal for modern VR headsets. Both for optics and weight and space reasons.