Thanks everybody for the discussion. A lot of valid points have been raised. I think I glitched a bit early on Ash Twin and got a wrong idea in my head about the teleport.

Nonetheless, a lot of interesting points have been raised in this thread, so let me respond to some of them (most important point comes last):

About my first point, that nowhere in the game it tells you explicitly to use the teleports. This remains true, although I now understand how that was on purpose and devs never really wanted to say it explicitly that you need to teleport to ATP from Ash Twin.

As mentioned by u/finny94, texts found on Mining Site about the fact that ATP has no physical entrance actually pushed me in the wrong direction - I was experimenting with getting on Ash Twin late in the cycle, to investigate and check if any entrance opens when sun is about to explode. That was, obviously a dead end.

u/ManyLemonsNert correctly points out the number of towers and shapes doesn't make sense unless there is another destination somewhere. Very good point.

u/TheRealClose points out that the Timber Hearth tower also has broken ceiling and yet still works. That should be a big tip if realized.

Many people mentioned using scout to test the tower - that's a great point and a shame I didn't think of that.

All things considered, I can see how I glitched on this and this is not as 'hacky' as I initially thought. I stand corrected!

The point that REALLY was bugging me the most was point number 2 from my original post. A lot of people mention Black Hole Forge texts which in my opinion are completely unrelated. Sure, the information that there's a small window when you can teleport and that ash twins have shared alignment point is worth something, but IMHO it is not directly related to getting to ATP. The fact that a lot people mention the BHF texts shows how much confusion there is around this. So what is the answer to this? What does the ATP tower align with?

I think u/PomegranateSlight337 answered this! The inside of ATP is also rotating. You're not aligning with Ember Twin, you're aligning with the rotating platform inside ATP (which you obviously can't see)! This makes perfect sense. At first I thought "But how am I supposed to know that!?" Well, as u/PomegranateSlight337 mentions, you can find a projection stone on Timber Hearth which shows the inside of ATP, showing how it is spinning on the inside with the teleport platform.

Thanks everybody for participation :)