I did something similar with BJT's a while back when I had no IC chips. I remember knowing I could add more buttons, but that it would take exponentially more BJT's. I am hoping my friend still has the schematic for it (I think I put it in the project box everything was built into).
It looks like you have a couple of flip flops built from transistors (I base that on the general layout of the upper corners, particularly the crossed resistors. If that's the case, then you have a storage unit that it makes sense to reset.
But the TS just has fixed switches and no storage element of any kind. Whatever 'reset' he uses to force the result to zero is going to go right back to the result dictated by the switches as soon as he relaxed it.
This is not too unlike the old mechanical voting machines that I remember from when I was a kid. When they stepped in they moved this big lever handle from one side to the other to close the curtain to give them privacy. Then they voted by positioning switches to point to their choice in each election. They could change it as much as they wanted to. When they were done, the moved the big lever back to open the curtain to leave. But that big lever also added their votes in each race to the running total kept in a mechanical counter and it also reset all of the switches back to their "no vote" position.
Instead of using a toggle switch you can alter the design to use a push-button.
When a voter presses the button the vote is latched into a SET/RESET flip-flop. Use the RESET function to reset all the flip-flops.