Does time really slow down as you approach the speed of light?

Time is measured by clocks. It is just the movement of matter. If a clock accelerates away from you the intervals between signals lengthens. It looks like the clocks passage of time is slowing down, but it is not, the signal is just delayed and stretched out.

At light speed the signal would never reach you and time would appear to stop. If the clock accelerates toward you the signals arrive at shorter intervals and it appears time is travelling faster.

Imagine a clock 1 light year away accelerating towards you just under the speed of light. It takes just over a year to reach you from the clocks perspective. From your perspective you get all the signals just before the clock arrives and it looks like the time for the clock is passing fast, but its just the signals have been compressed.

Can anyone tell me why this is obviously wrong?

Could you not travel faster than light? If you accelerate faster than light your light signals would just never reach home if your travelling away or would arrive after you left all at once. (Could Cherenkov radiation be something like this?)

Anyway I’m interested in learning why this is wrong. I just like to keep an open mind and question stuff.