Interesting. I've never heard of David Thompson's technique,maybe the part of the country where one lives would determine how well his method would work. I have heard several recommendations about storing tubers in the fridge, so that is not anything new. I store species bulbils in the fridge and they do okay,but I haven't room for tubers.

I am not an expert but carried over my begonias in the closet with the caladiums. I stop watering them in fall but they seem to keep going for a while. If they don't start winding down I dig them, leave the big stem on and stick the whole thing in my garage. At some point they separate easily from the stem. Don't leave any stem on them because that will rot. I knock off the dirt, don't wash. Then I use the paper from the shredder to separate them and put them in a fruit box and into the closet. Every body made it through last winter. Cool and above freezing or just dry will keep begonias dormant. An unheated garage that doesn't freeze is a good place.
Picking off the blooms and growth tips sounds like a good idea if you can stomach it. I hate to destroy a big pot of wonderful to start shutting things down for winter. I have 70 begonias this year so I don't plan to use the refrigerator to store them. I wouldn't use a plastic bag for storing anything. Dry and dark and above freezing works.