A while ago I asked how an Occupational Therapist would deal with an individual who didn't want to "touch" the activity. I think I found it...
In this case in the video, the guy didn't want to touch the shaver. He has autism but sensitivity happens in different disorders. Desensitization is used. ("Touch the shaver five times.") It's a variation of exposure therapy used in psychology and behavior therapy.
The electric shaver is great for independence. I personally use one to shave my legs. Nicks and cuts from a blade are not to worry.
There isn't necessarily the same kind of shave as a razor. However, shaving more often is worth the independence.