If you scrub an area with Betadine you're going to have to remove the Betadine before tape will stick. Or at least you will if you want it to stick for any period of time.

Try this experiment: Wipe some Betadine on your arm. Let it dry. Apply a band-aid over top of the dried Betadine. Betadine dries slightly tacky so it might feel like it is stuck on pretty well. Now try a little light exercise. Get out of the AC. Walk around the block a few times. Let it ride the rest of the day.

In my experience the dried Betadine absorbs sweat and softens. Band-aids and tape tend to fall off in a short time.

This may, or may not, be a consideration. If you're bandaging people up just to get them to a hospital a few minutes away it really doesn't matter what you do. You want to make a professional job of it but it is mainly out of pride instead of necessity. In short order the docs will remove the bandage. A carefully made field dressing is wasted if they are on their way to an ER or trauma center.

On the other hand, in a disaster, or austere field conditions, and certainly for minor injuries that won't see a doctor, wounds that account for the majority of what a FAK is used for, people will be wearing that bandage for a day or more. If you don't want to blow the contents of your FAK redoing one small injury you need to get it right the first time.

It also has to be noted that that time honored stickum, tincture of benzoin, is not entirely foolproof. I happen to love the smell. But I'm also mildly allergic to it. Anywhere it dries on my skin blisters in 24 hours. I had it used on me to anchor the ends of three Steri-strips. Next day I had six tight and round little blisters located directly under the ends of the strips. The laceration healed long before the six spots stopped their cycles of cracking and peeling. I now avoid the stuff.