In my opinion, Ek = mv² contradicts special relativity. The maximum energy is E = mc². There can be no doubt about it. And it is not rest energy. It is obviously kinetic energy. But that's just me.
The energy is not a primitive physical quantity. It is a quantity that is defined.
The definition of energy is not unique, it can contain an arbitrary multiplicative factor without changing anything in physics, as long as the same definition is used everywhere.
If the kinetic energy is defined as mv^2, everything is fine, but then the mechanical work is 2Fd and the total energy from the special relativity is 2mc^2.
If the kinetic energy is defined in the usual way, all the values of energy or work are halved.
There is nothing mysterious about this. It is just a matter of convention.
In the beginning, the kinetic energy was defined as mv^2, because it was the simplest formula.
After more complex mechanical problems began to be studied and solved, requiring differential and integral calculus, the definition was modified to be that from today, because there are more formulas that become simpler than formulas that become more complex, like the direct computation of the kinetic energy from mass and velocity.
The same has happened with the Coulomb law. Initially the electric force was defined as the product of charges divided by the square of the radius, because that was the simplest possible formula.
Later, it was realized that making the Coulomb formula more complex, by defining the force as the product of charges divided by the area of the sphere centered on one charge, many other formulas become simpler, so the old definition was changed.
The change of the kinetic energy formula was a good change, because computing a half of mv^2 is a negligible complication, while a large number of formulas containing integrals or derivatives become simpler.