Hi bronze, I have taught plenty of children how to ride their bike as it is my job. As ItsAllGoingToBeFine says the first point is to remove the pedals so they can learn how to balance. However this is not completely necessary as it can be a difficult job and you need the tools to do it.
I wouldn't advice to ride on the grass as it is more difficult to pedal and they will have to do extra effort ina moment where they are not ready.
However I would have them riding in a flat surface, maybe in a path in the park, while you hold their saddle from behind. They will be able to feel your hand which will give them confidence to try while they pedal and you run by their side. I would tell them to focus in a point in front of them (maybe a tree, bench, sign, ice-cream van
...) and not on the floor. This is because it is easy to keep your balance if your sight is focus somewhere in front of you and not on the floor or front wheel.
When you notice that they are able to balance themselves start removing your hand from the saddle for short times but still running by their side and be ready to put your hand in the saddle if they seem to lose balance. It is important that you keep reasuring them how well they are doing as a lot of praise will help them to increase their confidence. As you keep doing this make the time you hold their saddle smaller and stop running by their side, just a bit behind but not too far away so you can stop them if they are going to fall, keep talking to them so they know you are till there even though they may not see you. It may take a bit of time if their confidence is as low as you say and I wouldn't expected to be able to do it in one afternoon. Also try not to force them and wait untill they feel ready to try again.
It is also important that they are able to stop using the brakes and not their feet as it will make them falling more easily. Also it is important that the saddle is in the right high for them. They should be able to reach the floor when they are sitting in the saddle and their knees should be straight when their foot is in the lower pedal. If the saddle is to low it is more difficult to keep balance and falling occur more easily. This is something that as a parents we think it is a good idea (for them to be able to reach the floor easily) but it is a lot harder for them to keep balance.
For your DS1, have you considered getting him a tricycle? They are very useful and may solve your problem or you may want to get a tandem as the front der will have to keep all the balance. Also you can try and get some professional help. Sometimes it is easy if someone from outside the family can teach them as they will listen more and they will try harder. Depending on where you live there are help to pay for cycling training and even in some cases you can get free lessons. You can google cycling instructors and get plenty of information. Also most of the schools do "bikeability" which used to be "cycling proficency" for year 5 and year 6 for free depending on where you live. If it is the case in your children's school, you may want to get in touch with the company that offers the training as they will have instructors available to teach them outside school hours.
Hope this helps and good luck. It is fantastic to see their faces when they realise that they can ride their bikes on their own. I love it 