That's where you go off the rails again. Where does the "logic of the line" state that?
In real world terms this means I must wait for infinite time to pass first before my place on the line shows up.
Could you provide an example from the "real world" where someone has to wait for any amount of time before showing up on a timeline? I think you're really stretching the limits of comprehension now.
The line has a logic. It's logic states that any point on it has infinite time that has already passed before it.
So we TRY TO apply this logic to the real world.
I want to place myself on a real world point on this line but I must wait for infinite time to pass first. I can never do it.
The logic of the line prevents me from placing myself anywhere in the real world since that would require an infinite amount of time to pass first.
Only if you first make the assumption that time had a beginning. In the event of infinite time (no beginning) then there was an infinite time before that point for everything to have happened in already so there is no waiting.
This doesn't address the problem.
It is easy to imagine a line that moves infinitely into two directions. That is not a problem, nor a solution.
But if I want to place myself into the real world and also follow the logic of the time line I can't do it.
To follow the logic of the timeline means that to place myself at any moment in time requires an infinite amount of time to pass first.
So I wait a billion years to place myself on the line, but I can't do it. I have to wait an infinite amount of years. I wait a billion billion years and still can't do it.
There is no number of years I have to wait until I can place myself into the real world. The amount is infinite. I can never do it.