Special Relativity tells us - the faster things travel their time is slower relative to a stationary observer. Do massless particles, like photons travelling at the speed of light, “experience” zero time and in their frame of reference travel "instantaneously". Putting it in a thought experiment. Imagine travelling to a distant star. On reaching the star, several decades have past as shown on the spaceship's clocks and on our aging bodies. Now consider doing the same journey but incredibly close to the speed of light (not possible in practice but not violating any physical laws) we would see the clocks have changes only by a few minutes and our bodies are unchanged. Wouldn't it be reasonable to extrapolate and predict that if we were to travel at a smidgen less than speed of light, it would appear (to us astronauts) to have taken close to zero time?
Your reasoning is true. If an observer on, for example, earth were to measure the time passing on the spaceship, they would find that the time that passed for the astronaut is
where $t'$ is the time elapsed on the spaceship moving with relative velocity $v$ and $t$ is the time that has passed on earth. One can see as $v\to c$, $t'\to 0$.
One could argue that since for a photon, $v=c$, $t=0$. However, since $c$ must always be constant, there is no frame of reference moving along with a photon $-$ if there were one, the photon would be at rest in it, which would violate the constant speed of light principle.
So it does not really make sense to talk about how much time has elapsed for a photon or a hypothetical observer at the speed of light; but one can accurately describe what happens as the velocity of the observer approaches $c$, where the reasoning that you elaborated on is correct.