What should I look for in the rider?
Do the procedure below and just watch what happens.
Are there any activities that we could do to build confidence?
Start out with pedal-less bikes (walking bikes). No danger at all, plenty of time to have fun. This can start at toddler age. They will learn all the balancing on their own, when they are ready. When you notice them taking a run and then sailing along without crashing at the end, you know that they are more than ready.
By the time they are strong enough to turn the pedals of the abysmally heavy children bikes (relatively speaking), I grabbed my kids (individually) and took one very long ride with them. I walked along; at first I held them very strongly (i.e., grabbed both handle bars). After a few 100 meters, I held them at the hips and let them try to steer, but still holding strongly enough that I could just pick them up together with the bike, if necessary. After a few more 100 meters, I only vaguely stabilized them. At the end of this (maybe 2-3km, on a very nice sunny day, with obviously them not pedalling all the time, but me pushing/holding as necessary, they had it pat down and could ride their bike on their own.
At the beginning, you are 100% in control, and they have 0% control. At the end, you have 0% and they 100%. You can stop at any time if you see it does not make sense.
I had great success with this "one-piece" education. For whatever reason it worked out - this is a test sample of size 2, not a consensus, but maybe food for thought.
If, at the beginning, you notice that it simply does not work out, then just stop. They don't even have to know what you had planned (no hard feelings etc.).
Should I remove both training wheels at once, or one at a time?
Skip wheels. Wheels are always bad. Bikes are stabilized by speed. It makes no sense to stabilize a standing bike. If they are not able to develop a certain minimum speed (at which point the bike will be stable), and hold the handlebar straight, then they have no benefit of sitting on a stabilized, standing bike. They will only learn that they can lean to the side and nothing happens - this is a very detrimental behaviour on a moving, training-wheel-less bike.