I don't understand energy

Acceleration is proportional to force, velocity is the integral of acceleration over time, and displacement is the integral of velocity over time. That's a nice symmetry of the universe. The displacement of an object is just the second integral of the force over the mass with respect to time.

\(\displaystyle \Delta position = \int \int \frac{\sum F}{m} dt dt \)

If this symmetry continued forever, then kinetic energy would be the integral of force over time. If I apply 1 newton of force to an object for 1 second, then logically I should add 1 joule. Unfortunately, it doesn't work that way. Instead, energy is the integral of force over distance. If I continuously apply 1 newton of force to move an object 1 meter, then I've applied 1 joule of energy, even though I've applied that force for \(\sqrt{2}\) seconds.

One obvious solution to this problem is to think of "energy" as a purely human invention. If I want to move a block from point A to point B, it's easier to have a number that works on distance rather than time. There is, in fact, another unit for force integrated over time; it's called impulse.

That doesn't work though, because there's another definition of energy that I haven't mentioned. If I take 1 mole (\(6.022 \cdot 10^{23}\) molecules) of \(H_2\) and 2 moles of \(O_2\), and react them to form 2 moles of \(H_2O\), 572 kJ of energy will be released. If I instead take 2 moles of \(H_2\) and react it with 4 moles of \(O_2\) to form 4 moles of \(H_2O\), it will release 1144 kJ of energy.

The "enthalpy change", or change in energy of a reaction is directly proportional to the number of particles reacted. It's not at all obvious to me why particles should move at a velocity proportional to the square root of their kinetic energy (temperature), rather than directly proportional to their kinetic energy.

I really don't have an answer to this. I took AP Chemistry last year and I'm taking AP Physics this year, and something just doesn't click for me.