Why a Conspiracy?

John Titor, take note.

When you are stood or sat in your basement, lounge or wherever and your mind springs to time travelling; you have to understand a few things.

 Time only exists on a time piece, whether that's a watch or clock or sun dial, etc. You know, things like that. Time is merely a saying and not a doing word. You cannot go back or advance physically by manipulating the pointers of a watch or clock or deflecting a shadow of a sun dial, because all you are doing is simply counter rotating or advancing the pointers to man made numbers. You do this is real time.

What is real time?
It's the time that you do anything based on following a man made timepiece that is used to make people follow a pattern, in the "NOW."

If you record someone on a video on a set date and time in what you regard as your past, you record them in your present time.
If you want to view them as you advance in age, you can replay the footage in your present time but you cannot look back "physically" in time. You can look at a video in the present of something you did in the present of the time and date that was.

You cannot crawl back into your video and go and be a physical part of that time because it isn't in the past, it simply has PASSED as a memory that is recalled on a separate mechanical memory, as well as many other organic memories, probably.

Like I said earlier. It's good to think of stuff like this and it's great for sci-fi  thoughts and films but it's not for our reality and never will be.
We humans think we are special on this Earth. We are no more special than a cell under a microscope. We are only as intelligent as we arrogantly believe ourselves to be.
We are intelligent enough to adapt to use our hands and minds to aid our survival. We advance in deterioration from birth to death like every other organism.

We can't stop this from happening. We can arrest it for  so long but we cannot change the world we live in, in terms of going back to other times and dates, nor can we do it forward.
The Earthly matter knows only to do what it does, like it's always done, in whatever way it's been done.
No wires or magnets, or dials or digital numbers will do anything other than break down our cells to hasten our demise. It will not change our perspective on living in a past life nor a future one.

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Time travel is impossible, but I will play along for the hell of it.

Why do you think that?

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Time doesn't exist. You create your own fictional imaginary reference point.

We do create reference points, but to do so there must be something to refer to. Time is just another direction.

You can create  reference points whilst walking through a forest by dropping pebbles, then retrace your steps to get back to the start.
You haven't advanced in time nor went back in time. Everything you do, is done in your present time; always.
You may walk forward to drop your pebbles, then pick up those pebbles as you about turn.
All you're doing is retracing your steps in present time. You pick up your very first pebble you put down as you come back to the start. You do this in your present.

A reference point is simply just that. It's a reference point to enable you to retrace your steps. No time travel happening, nor ever will.

You haven't advanced in time nor went back in time. Everything you do, is done in your present time; always.
You may walk forward to drop your pebbles, then pick up those pebbles as you about turn.
All you're doing is retracing your steps in present time. You pick up your very first pebble you put down as you come back to the start. You do this in your present.

I agree, everything exists in my present: however, what is my present may not have always been the same as yours. A month ago, by my perspective, my present was what you'd call your future.
Time travel is travel through the time of another's reference frame. From my perspective, I simply am. If I take a certain event however, I will clearly exist in the past or futue or present, relative to  that event. That is all time travel is. If we choose an event that occurred at the same time as me entering my time machine, while that event would once have been by present, it is no longer.

All time travel is, is movement in a new direction.

It is interesting you think of it in those terms, however. One of the forerunners to modern time travel was the D-mail: essentially, a way to send emails and texts to a device in the chronological past. The only way to tell they had been sent, was to see that the phone responsible has no record of actually sending the mail. Remembering the past worldline would, obviously, be impossible (at least, theoretically. A friend of my father's designed the D-mail before I was born, he claimed to be able to remember the past worldline. Then again, he always was rather pretentious, it may have been a lie).
If you believe him, however, when the message was sent, he didn't remember anything of the new worldline. He was simply thrust into the same point in time, resulting from whatever past was altered by the message. However, there was no true time travel involved: essentially, only the present was altered.

Many of your objections do seem based in a misunderstanding. I am the D-mail, in this case. I am irrelevant: what matters is what I call my present, your future. Everything there will shift: the present will change. The details here are what leads up to and causes that to happen, but depending on your reference frame, only one of two things is happening (relative to the worldline). I am someone who came out of nowhere, and am simply part of events: or everything that happens now is no more than a memory, and my present, your future, has simply shifted worldlines.

I hope this was clear. The theory of time travel is still not completely understood, but this is what I know.