Why should signals in circuits propagate much slower than the speed of light?
think about it like this, most electrical signals travel at the same speed(depending on the material)

imagine we have a 60Hz sine wave, the wavelength of that is around 6000 kilometers

the wavelength of your signal is 10 cms

The 60Hz signal is not slower, the wavelength is much longer, I think this is the root of your confusion.

for lumped circuits you want to ensure that given a distance all the values of the signal stay the same, they change over time, but not over distance(they do but it is negligible)

take this schematic, imagine the conductors that connect the resistor to the other components, in your example the wavelength is 10cms right? so in 10cms distance it is a whole period so if the wave starts at 0 goes up to maximum, back to 0 then negative and back to 0 again in the span of 10cm. So what length can you ensure  that the values stay the same in the whole circuit? for example the whole circuit is at 0V at a given instant... there are no electrical signals on the way or any point in the circuit waiting to get the memo.

for 60Hz I think most people stop using lumped circuits at a couple of hundred of kilometers, because already at that point there is "travel" delay and reflections and everything else make a significant difference along the conductor.

for a signal like the one you mentioned, even at 1cm the singal changed a 10th of what it would on a full period correct? 







simulate this circuit &ndash; Schematic created using CircuitLab