Here's the thing: Ted was supposed to die.
He has literally seen what happens when he doesn't. Saying that the world goes to hell in a handbasket is putting it lightly. And in the grand scheme of things, Ted's not that important. A few people will mourn him - Booster will be miserable, if this rescue attempt was anything to go by - but someone else will be inspired by him. That kid, Jaime.
Ted inspires someone, the way Dan inspired him. It's a heady, incredible thought.
And it's one of two things that get him into the time sphere, knowing he's going to his death, with Booster shouting at him from the other side of the glass. His other motivation - seeing Booster fade away in front of him, another in a long line of deaths Ted being alive caused - is what makes him slam on the launch button as fast as he can. If he lives much longer, Booster as Ted knows him is going to disappear.
For all he knows, he's too late and Booster doesn't even exist anymore.
He manages to put aside the heartache and fear once he's in the time stream. That... is what Booster called it, isn't it? God, he is so out of his comfort zone here. He searches everywhere for a manual, despite a feeling in his gut telling him Rip Hunter wouldn't have one - the risk of an unknown getting their hands on one and flying off into spacetime is too high. Sure enough, Ted doesn't find one, which means he's playing this by ear. It looks like he can program a destination based on local time and coordinates, which Ted is decently confident he remembers. He enters them as fast as he can, pulls up a preview image to make sure, and - whoa.
He's got the coordinates right, and the time is just a bit late - it's after Booster and the Beetles charged in, knocking Max out and saving him. But... no, now it's - Ted forces down a bout of nausea - it's Max shooting him in the head. Now Booster again, now Max, now Max sitting at his desk like Ted had never come in, now Booster but the Beetles he's got with him look different, now Max, now -
An alarm starts blaring, and a box full of big, bold words starts blinking on the head-up display, but Ted can't read the language, though he recognizes it - Esperanto, Booster's native tongue. Outside, the timestream has gone dark and bold, the opposite of its usual pastel shimmer, and a shudder runs through the time sphere. Ted tries to hold on as best he can, but the shuddering gets worse, and he loses his grip, head banging against a particularly unforgiving piece of equipment.
Dimly, he hopes that he hasn't done Max's job for him, and slumps against the console, insensate.
When he wakes up, the time sphere has landed. Crashed, really; the thing is a wreck. He crawls out and looks around, worried that he had to see that look on Booster's face and still condemned the world to that hell-hole - but no, everything looks... fine. The sky is dark and starry, the grass at his feet looks green enough, and he thinks he recognizes the Chicago skyline in the distance. And in the other direction - Ted stares.
It's Kord Industries. What are the odds?
Since he's already here, it just makes sense to head for the Bug cave. He remembers where the security cameras were, and hopes there haven't been any big upgrades since his death. Assuming the ship landed some time after his death. No one catches him before he finds the right door, and with a touch of his hand and an impulsive "Open sesame," he's in.
Everything's where he left it - the Bug, old equipment, row upon row of slightly outdated computers, and there at the end is the one he wants to see. The entry log. He checks it.
Today is nearly two years after his death. And here he is, not actually dead - and the world is fine.
Somehow, Ted's survived without dooming everyone.
He bursts into joyful, slightly hysterical laughter.