Trying to understand clock circuit (wall clock, not signaling clock)
The driver in solenoid clock is a peculiar and almost unique circuit. As in, it's used only in clocks but there are a lot of these around!
The blob is a dedicated IC, you most probably see a tiny little metal can near it: that's a 32.768kHz crystal. 32768 is 2 to the 16 power so with a simple binary divider you can have a 1Hz signal. On each of these tick the ICs gives a little pulse to the solenoid and advances the clock (there are variants, some clocks go a 2Hz due to the mechanism, for example).
The trick in that circuit is that its consumption is in the microampere range, so a battery can drive it for years. I think it would be quite difficult to redo it as a discrete circuit and I don't know if such an IC is available in packaged form
To have an idea of how modern clocks could work read this http://blaauw.engin.umich.edu/wp-content/uploads/sites/342/2017/11/YoonA558nWCrystalOscillatorUsingPulsed.pdf