We derive length contraction from time dilation. Here, we are adding these two together. Aren't we double counting here?

Both effects are in play and have to be counted.

Looking at the video it looks like the clocks at x=0 and x'=0 are in sync and stay in sync. Shouldn't the clock at x'=0 start to lag more and more in respect to the clock at x=0 due to the time dilation? All seen from the standpoint of team platform.

Clocks in motion run slow, objects in motion are Lorentz contracted, time is dilated. All this was ok. It has taken me several days to absorb the maths supporting the concept that clocks moving away from the observer run slow whilst those moving toward the observer run fast - my brain must be running away from me!

fascinating fun to imagine it thanks brain greene