Really!? It never ocurred to you that if all you had to do was increase the clock frequency to make a processor run faster, that someone would have done it by now. You actually think that Intel and others with 1000s of engineers looking at this problem didn't realize that all they had to do was increase clock frequency instead of spending a few 100 G$ on a new fab? Duh!
No, it doesn't work that way, obviously.
First, electrons actually travel remarkably slowly in electrical conductors. However, that doesn't matter. It's the propagation speed that matters.
Second, signal propagation speed is not the limiting factor in modern processors. Stop and actually think about it. What's the speed of light? Even figuring signal propagation speed is half that due to the impedance of the transmission line, how long would it take a signal to cross the die? No, actually go figure it out.
The real limits in today's leading edge digital logic come from having to charge and discharge the inevitable capacitance on any conductor you are trying to switch the voltage of, and the reaction time of the semiconductors. Both those cause delay from starting to drive the input of a gate until the output reaches the threshold where downstream circuitry will reliably interpret it as the intended high or low level. Each individual gate may be quite quick, delaying less than a ns. However, to do meaningful things in a processor can a number of successive stages of gates.
Much of high performance processor design is reducing the gate delays in the worst case path. Often shorter delay can be traded off with complexity.
For example, look at a basic adder. Each stage takes in the two bits to be added, the carry from the previous stage, and produces the output bit and the carry that is passed onto the next stage. In a basic adder, the gate delay therefore grows with the number of bits. Higher bits can't be added until the carry from lower bits is available. This basic adder is also called a ripple carry adder. There are other types of adders that have lookahead carry. These take more gates, but can add two wide numbers faster. More gates of course means more cost and more power consumption, which means more cooling required, etc. Nothing is free. This same principle of more gates to make things faster applies in many places.
Then there are other things, like memory, which sometimes work on a different principle than gates, and also have their inherent delay times.
Back to the original question, the point is that these gates all function at some minimum delay from when the inputs are stable until the outputs are guaranteed to be correct. One way this is guaranteed is by latching the inputs on one clock edge, then not using the outputs until a subsequent clock edge. The people that design the processor very carefully decide how fast it can be clocked over what temperature and voltage range so that all the gates have the right answers by the time those answers are used.
Another limiting factor is being able to get rid of the heat. All those little parasitic capacitances being charged and discharged causes current proportional to the charging and discharging frequency. This means faster clocks cause higher current, which causes more heat, which has to be gotten rid of safely so that the chip still functions. Silicon stops being a semiconductor at around 150°C, and of course you need some margin below that. If the gates are packed too closely, clocking them too fast would cause them to get too hot to function within your capability to remove the heat. This is why in some cases you can overclock some processors by cooling them more than intended. Note that this only addresses one of the limiting factors, so you can't keep overclocking a processor no matter how much you are able to cool it.
Anyway, the maximum clock rate of any processor is a complex subject. No you can't just clock it faster to make it run faster and still have it work, and no, the limiting factor has little to do with how fast electrons travel.