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Not knowing how widespread this apporach may be, I'll just contribute, what strategy my parents applied:

My father bent the trainers more outward/upward - this left more room to tilt from one side to the other, or to balance in between. I figured that it would be cool (and less noisy) to try to keep both trainers off the ground. Soon afterwards, the trainers were gone.

Or, you can just wait until love makes the trainers obsolete:

The accepted answer is correct: don't use training wheels. Once my kids got interested in bikes, I assembled the ones I bought for them and just left the pedals off. After a couple sessions of learning to push themselves around on flat pavement and gentle slopes, they could push off with both feet and coast while balancing. They were ready. I put the pedals on, and they immediately started riding. No falls, no scraped knees, just movement.

The only fall that day came after a great deal of riding. The oldest overcorrected on sharp corner at top speed. Aggressiveness coupled with inexperience. Somewhat inevitable, but at least my helmet investment paid off and saved a lot of damage to a forehead.

Your first step is to get a bike that's the right size. Being able to touch the ground while sitting in the seat is essential. The other thing is the bike's weight. The smaller and/or less muscular the child, the lighter the bike needs to be. It really makes a difference. For more details, this article seems to cover things nicely. Finally, try to find a paved surface that's in good condition. We wasted some time in a rundown parking lot nearby and had much better, faster results when we picked a better location.

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.

IMO the condition is to have an adult available to help the kids to build balance skills.
The time needed is surprisingly short.
It need not be an adult, it can be a teenager.
Do the training on a soft leveled ground. Grass is the best.
Ensure that the bikes don't have any prominences that might harm the kids.
And, as Carel said, lower the saddle.
And, very important, greet them as they progress.

Training wheels are good if they have not learned to pedal. Once that is done start moving the training wheels off the ground, a little at a time. If the kids are nervous don't let them know you are doing it, they will adjust to it without thinking about it. If they have pedaling down and are the braver sort. just take the training wheels off, as others said speed will keep them up. If they are older you can explain the science of it by taking the front wheel off and spinning it. It will stay up stay up with holding just one side, like a gyroscope.

In our family, one of the secondary benefits of training wheels is that it allowed the youngest to bike with the family.

I secretly raised the wheels a bit at a time until we told him to see how far he could go without either one touching.

Once the bicycle is moving forward, the training wheels are unlikely to be touching the ground. Their main function is stability when starting and stopping. Video the child riding to demonstrate this. Then wait until the message sinks in and they're ready to ask you to remove the wheels.

Forget the training wheels. On level ground, preferably no pavement & definately no gravel, push & steady the bike. Your child will learn very quickly and you'll know when to let go. It took me about 3 hours to learn this way.