I think you misunderstand your reference. The line "Assume tachyons exist and special relativity is correct" is a logical contradiction, and I will attempt to explain why.
The thought experiment you linked requires a logical violation which any person of reason knows to be impossible (specifically, greater than infinite energy.) It is not predicted by relativity, it is a discussion of what would happen if you tried to apply it in a situation to which you cannot apply it.
Einstein did a lot of this kind of thing (it was probably fun for him to discuss it.) It is not part of his theories on relativity, and you should probably look at it more as a discussion of the consequences of applying his theories in situations they are not meant to cover.
You are assuming greater than infinite energy AND time travel to even put it forward as an argument...
I see the greater than infinite energy requirement as a hard physical limit to the theory, and an implied limit for the equations involved. While this limit is not explicit in relativity, I think we can safely assume that Einstein did not intend for the theory to apply in a situation which requires greater than infinite energy.
As you approach the speed of light, energy input must approach infinity. To cross it requires a greater than infinite amount of energy.
This is impossible, both in theory and practice. This is not just a "we do not know how" situation, it is absolutely impossible within relativity.
If you do not understand why greater than infinite is absolutely impossible... I am not sure what else I can do to explain it.
He was aware that infinite + 1 does not really exist, you are apparently not. It is a "what if?" not a prediction.
Relativity is a physical theory, if you take it beyond this point you are talking about a purely mathematical construct. You cannot apply this to the physical world, as infinite + 1 does not exist here.
Relativity therefore prohibits producing a tachyon. It is impossible according to the theory due to the greater than infinite requirement, not the causality violation which this inappropriate input would produce.
Trying to do so anyway makes me think of the saying "garbage in, garbage out."
Everyone who initially discussed that was aware that it is an impossible scenario when they did so, which is why they assumed it for the purposes of that argument. They determined that the already impossible scenario would also produce causality violations, which means relativity breaks down here. This does not detract from the theory, as you cannot get this result with any valid input.
It is in fact theoretically impossible to produce a tachyon within relativity. Absolutely impossible.
If FTL communication is discovered at some point we are no longer living in a world ruled by relativity, and this must be accounted for. The theory prohibits this, and relativity has no appropriate application in an FTL scenario (this is WHY you get a causality violation.)
Causality is not relativity, it is merely that cause precedes effect. I am saying that while your observation of cause and effect may not be correct, it still does not allow you to change the past.
The cause comes before the effect, despite your observation of the order of events.
Let us construct a thought experiment of our own by taking a trip to sci-fi land:
Two advanced alien species are at war, and they have the capability to move objects from one point to another in the universe without crossing the space separating these two points (FTL.)
Side A fires an FTL missile which destroys a ship on side B.
The observation in half of the relativistic light cone will be that effect preceded cause (the missile destroys the ship before it is launched.)
So what happens?
1) Causality violation means that side B can fire their own FTL missile, destroy the ship on side A that launched the first one, and "undestroy" the one on side B by changing the past.
2) Causality remaining intact means that while side B can fire their own FTL missile and destroy the side A ship, the side B ship will remain destroyed, as this event has already happened.
I am stating that the observation of the order of events is inaccurate due to the introduction of FTL, and the second scenario is the correct one. Cause still comes before effect despite the fact that the order in which you observe events does not show this.
The information on the cause is relativistic, the information on the effect is not. You cannot use relativity to determine cause and effect here, but cause still came before effect.