Wouldn't it be easier to just say "current is how many electrons flow through a specific section of the wire in a second" and "voltage is the force that makes the electrons move"?

It's just as easy to understand without need for analogies.

Those two sentences that you wrote are precisely the hydraulic analogy. If it weren't, instead of electrons it would be charge (that can be positive or negative), or field strength (if you explain the transitory period of a capacitor or non-DC), and voltage would become not the force, but a number that applies to conservative fields that allows to calculate the capacity to do work relative to another such number...

Absolutely not. You can't just throw out axioms and expect people to catch on. The utility of analogies is that you can anchor an idea in something people are familiar with.

I would say that, in general, this is a tool most useful to help people that are completely new to the concept gain an intuition about the behavior.

Based on what you've described, where would you develop the intuition about the difference in speed between the electrons and the electromotive force they are relaying? Or why you need both voltage and current to do work, or the behavior of an LCR circuit? Or an open circuit? Etc etc.

Same, by the time you need to explain how transformers use electromagnetism, they probably understand the basics of V=IR which is really as far as the analogy goes.

Could a turbo-pump can be thought of as a very bad transformer? able to transform a flow mv1 at a pressure P1 to mv2 P2 where their product, power is constant (except its way more lossy).

Still, I think the water analogy is over used

That’s probably a good description of the main usefulness of the analogy - explaining to people who won’t actually need to apply the concepts. My conclusion (after years of doing electronics design professionally, including RF) is that to do much useful in electronics, you need to get into the habit of thinking natively in terms of voltage and current and resistance etc. so soon that the hydraulic analogy is basically not worth learning at all, since the missing things just get confusing and it becomes a distraction or blocker to advancement until you “unlearn” many of the concepts.

I mean, it’s probably worth mentioning current and voltage are “kind of” like flow, pressure etc., in a first lecture to start to get the basic idea, but then warning not to think of it as being the exactly same because that will just be confusing later.

You can do 99.999% of practical applications of electricity with the liquid analogy, up to and including most industrial electrical stuff (remember, check valves, accumulators and whatnot are a thing).

What you can't do is design a bottom dollar power supply that uses some trick circuitry to not emit enough RF noise to matter.

I'm specifically talking electronics, where the usefulness runs out very quickly... As soon as you get to EMI/EMC there is no real liquid analogy (so that rules out basically any reasonably complex custom electronic design you're going to get certified to sell as a product). Parasitic effects of traces, ESR, characteristic impedance, etc. are similarly problematic (people try to come up with stuff like springs or flexible tubes for characteristic impedance but again it's just more confusing in practice). So that rules out anything decently high speed digital (like, say, routing DDR3/4). That's of course not to mention RF, antennas, transmission lines etc. where all bets are off.

>> 99.999% of practical applications of electricity

All radio/wifi and everything involving fiber optics or magnets. That's probably slightly more than 00.001% of applications.

Not still a good analogy, but radiation can be seen as a consequence of vacuum not being empty from the perspective of an electric field (it has a very low permittivity, but it is not zero, so it is not empty of that property) even though it is empty of mass. If your water pipes are a bit porous and the medium on which your water pipes are was not air, but a very dense fluid, radiation emanating from the porous pipes would travel at the speed of sound through the medium, and could be detected without particles of water actually travelling through the dense fluid that constitutes the medium where your hydraulic circuit is submerged.

There's a large difference between "applying the concepts" and "doing useful things in electronics". The vast majority of people barely find utility in repairing what they already own, to say nothing of tinkering or creating something new.

The water analogy, is, in my opinion, the sweet spot for people who will never own a multimeter or fire up a SPICE program. It's the level that ought to be the expectation for citizens of a technic civilization.