Trouble getting 555 timer IC circuit to work properly (Make Electronics Book Experiment 18)
What if you put R10 to 10KOhm and inserts an 82KOhm instead of C4? The time between starting and the light switch is controlled by IC7, so the connection from IC7 pin 3 to IC6 pin 4 is only to turn on the light, there is probably used a 555 because it is easy, as IC6 really only works as a flip flop.

Update:

This is from a simulation. Hope it gives you an understanding of the signals.

0 = LED off

1 = LED on

S4 = enabled

S3 = enabled

Yellow = IC7 pin 3

Blue = IC6 pin 4

Red = IC6 pin 3

Green = IC7 pin 6/7