The largest problem is probably trying to sell binary code as a number system, particularly in this age group. As some other answers suggested the combinatorics aproach is probably the way to go. I would even stay away from 0s and 1s because that already has another meaning to them.

I also think that the beads is a good starter to first give them a grasp of how the number of possibilities in which to order them.
My idea would be a game where 5 kids are sitting in a row. This is basically the second phase after the beads. Each of the 5 kids has a number assigned in the order 16, 8, 4, 2, 1. Then the class is asked for a number between 0 and 31. If it is 21 the procedure goes like this. The first kid just has to check if his number fits into the target number. In this case it does and he stands up, the next one (or the class) has to now check the difference 21-16=5. Again check if the nummer fits, which it doesn't, and the kids remains seated and might say "done". The next on fits again and get's up, leaving a remainder of 1, kid no. 4 remains seated and kid no. 5 gets up for that remaining 1. This could be written to the board as 21 = +o+o+ or filled an empty circles or the class can be asked which symbols the would prefer. u and d might also be intuitive for up and down.
There can be a few examples, eventually well picked neighbouring numbers to show a flip of 2 adjacent positions.
If it goes well up to that point you can even count up, but as a teacher you should be familiar with that, so that it goes right and the system is applied correctly.
To finally close the bridge to a general application of this you can explain that the same can be done for the alphabet and instead of counting up numbers you can also count up letters.
It is possible that 31 is a bit high for 5 year olds for the calculation part. The good part is the physical involvement, which I think is a useful tool to enter their brains.

What also just came into my mind is Morse Code, which is also somehow binary in using short and long sounds. But I really have no idea if kids today know anything about it and if this would be a useful tool. Maybe a question to the class might give some feedback in that regard. It could also be mentioned on the side in a few explaining sentences.

The largest problem is probably trying to sell binary code as a number system, particularly in this age group. As some other answers suggested the combinatorics aproach is probably the way to go. I would even stay away from 0s and 1s because that already has another meaning to them.

I also think that the beads is a good starter to first give them a grasp of how the number of possibilities in which to order them.
My idea would be a game where 5 kids are sitting in a row. This is basically the second phase after the beads. Each of the 5 kids has a number assigned in the order 16, 8, 4, 2, 1. Then the class is asked for a number between 0 and 31. If it is 21 the procedure goes like this. The first kid just has to check if his number fits into the target number. In this case it does and he stands up, the next one (or the class) has to now check the difference 21-16=5. Again check if the nummer fits, which it doesn't, and the kids remains seated and might say "done". The next on fits again and get's up, leaving a remainder of 1, kid no. 4 remains seated and kid no. 5 gets up for that remaining 1. This could be written to the board as 21 = +o+o+ or filled an empty circles or the class can be asked which symbols they would prefer. u and d might also be intuitive for up and down.
There can be a few examples, eventually well picked neighbouring numbers to show a flip of 2 adjacent positions.
If it goes well up to that point you can even count up, but as a teacher you should be familiar with that, so that it goes right and the system is applied correctly. To explain what I mean it should go like this. Right kid gets up "one", first kid down and second up "two", first kid up "three", first and second down and third up "four" ... Imagine the power of for example 8 when 3 kids sit down and one gets up.
To finally close the bridge to a general application of this you can explain that the same can be done for the alphabet and instead of counting up numbers you can also count up letters.
It is possible that 31 is a bit high for 5 year olds for the calculation part. The good part is the physical involvement, which I think is a useful tool to enter their brains.

What also just came into my mind is Morse Code, which is also somehow binary in using short and long sounds. But I really have no idea if kids today know anything about it and if this would be a useful tool. Maybe a question to the class might give some feedback in that regard. It could also be mentioned on the side in a few explaining sentences.