Is the elevator analogy of the equivalence principle really true?

The point of the thought experiment isn't to say that the elevator can accelerate forever. The point is that acceleration is indistinguishable from being in a gravitational field. The acceleration doesn't have to exist for all time. However, just to address another issue...

It's limited by the speed of light. So at some point or another the elevator will start to slow down.

This is incorrect. The speed of light limit doesn't say you will start to slow down once you get closer to the speed of light. The issue is that you are thinking in terms of absolute velocity, not relative velocity. The elevator cannot accelerate to move faster than the speed of light relative to something else. This doesn't affect the acceleration you would feel in the elevator though. If we had some magical infinite fuel source, then indeed you would feel the same acceleration in the elevator forever. The speed limit applies to an outside observer who would see your speed relative to them approach but not reach the speed of light.

To begin with the equivalence principle claims that there is no difference between standing in a gravitational field and constant acceleration in deep space. This is only almost correct. The correction being that real gravitational fields have noticeable tidal forces (This is what made Einstein think of Curvature causing Gravitation). So if you were indeed in a real gravitational field, you can feel the tidal forces (if you are sensitive enough) stretching and squeezing you, while no form of constant acceleration can really make you feel tidal forces. However it must be noted that when Einstein gave the equivalence principle he clearly mentions that it is true locally, which means that there is no measurable tidal forces. But if you try to generalise the principle to larger scales, the ep is not valid.

Again by acceleration we mean four-vector acceleration, which is defined as $$a^\mu = \frac{d^2X^\mu}{d\tau^2}$$ Where $\tau$ is the proper time. This gives us the relativistic version of acceleration, hope that clears your doubt about acceleration as well.

The thought experiment is to show that gravity and acceleration are equal. It has nothing to do with how long an acceleration can be sustained. If that’s all your worried about just imagine the elevator swinging in a large arc where the centripetal force creates artificial gravity.