How does a car gain kinetic energy?

how does that energy become the kinetic energy of the car, since friction force from road doesn't do any work?

This is something I've seen several times on this site lately, and I disagree with it.

Static friction does do net translational work on the car. It applies a force in the direction of displacement; work is being done on the car due to the static friction force. I cannot see any way around this with the definition of work.

The engine (through the transmission) does work on the wheels. This is what causes them to spin. The spinning wheels are now able to do work against the road, and the road provides a nearly equal and opposite work back, with some losses. Since we are talking about the work done on the car, not on the car+road system, we can see that when you isolate the forces acting on the car, the static friction absolutely does work by the traditional definitions. It is providing a force in the direction of motion.

If we ignore friction (like your question mentions), the road obviously cannot do work on the car, and all the power from the engine just goes into rotational work of the wheels. You need the wheels to be coupled to the road by friction to actually get any translational work/kinetic energy from this rotation. This is how the static friction does work on the car.

There have been several answers given that address the main point that friction serves to convert the energy provided by the engine into kinetic energy of the car, but none seem to address the mechanism behind this transfer of energy. The only force accelerating the car along the road is static friction, seemingly indicating that the road is doing work on the car. If the engine is supposed to be supplying the energy, what gives? In particular, you've asked in comments "where would the road get [energy] from?"

Let's imagine the scenario in which there is no friction between the wheels and road. As the engine runs and the car remains stationary, the engine still delivers energy to the car-- in the form of rotational kinetic energy of the wheels. That is, without the mediating force of friction, the direct result of the engine's work is to provide rotational kinetic energy to the wheels.

Now let's switch on friction, so the car begins to accelerate. As noted before, we're forced to admit that friction is doing translational work on the car, being the only candidate force to provide it. However, that's not all friction is doing-- the static friction force is also imparting a torque on the wheels of the car in the opposite direction of their rotation. Recall that just as forces do work according to $\int \vec{F} \cdot d\vec{s}$, torques do work according to $\int \vec{\tau} \cdot d \vec{\theta}$. The observation to make is that if a wheel has a radius $R$, the no-slip condition of the wheel's rotation (i.e. the condition that the friction is static) is that $ds = R d\theta$ as the car moves a distance $ds$ and the wheel rotates through an angle $d\theta$. Since the torque and force due to friction on a given wheel are related by $\tau_f = R F_f$, we see that $$W_f^{tr} = \int F_f ds = \int F_f Rd\theta = \int \tau_f d\theta = -W_f^{rot}.$$

That is, the translational and rotational works done by friction are equal and opposite (the negative sign in the final equality is due to the torque's being opposite the rotation of the wheels), so that the total effect of friction is to do no work on the car. In this way, we reconcile the seemingly conflicting observations that the road transfers no energy to the car, yet it does the translational work accelerating it.

Flipping this statement around, we see that the work done by the wheels on the road is equal to the translational work done by friction on the car, suggesting the interpretation that the road "gets its energy" to accelerate the car from the wheels' rotational kinetic energy, which in turn was sourced from the engine.

Without friction your car won't move even a bit. Even though no net work is done by the friction but it acts as an energy converter and delivers the internal energy supplied by the engine to the car in the form of translational kinetic energy.