It didn't take long to find problems in the last diagram of Sequencer design
One is a stupid bug : the ring oscillator's output feeds itself and other relays. When the output relay is close, it feeds +Vcc to the capacitor and the other relays. But when it is open, the capacitor has more coils to drive and it empties faster...
A simple diode should solve this.
Now, this diode drops some voltage. I have tested the ring oscillator down to about 3.4V so 4.3V should be OK.
The clock output also feeds the pulse counter and the main CLK output. This last signal must be buffered through another relay, with the coil in series with the first counter's relay. I've added a lamp in series but I'm not sure it will actually change anything, or shine visibly. We'll see.
Fun fact : no freewheel diode is necessary because the overshoot will charge the ring oscillator's first capacitor. The changes are outlined in red.
The final counter stage is a more intricate problem, though.