I have been flying the US skies as a flight instructor and freight pilot. I have instructed students on F1 visas. Examiners just want proof that you're legal to stay here. Visa type does not specify pilot training in the first place. So, my assumption would be, yes, you can do flight training on your F1 as long as you maintain your F1 status at the university. If interested, I'll be glad to ask the FAA this question myself and post it here.