It's pretty much as you said. Non-bio is not as harsh for skin, and not quite as effective as getting stains out. But we always use non bio for whole family and it's fine. Just use vanish stick on stains before wash and you should be fine.
Having used non-bio for years I have now read (as a result of buying cloth nappies for the first time) that the enzymes in bio get used up in the first part of the cycle and in theory there shouldn't be anything left to irritate skin by the end of the wash. Started to use some, for clothes that are stained, and it hasn't irritated me at all (and I had to stop using the laundry service for bedding when I lived in halls of residence years ago because of terrible rashes) so I'm guessing bio powders / liquids have changed alot in the last 20 years.