tl;dr - There may not be a Barry Allen on Earth-X, and he may have never gotten powers if there were.
While there are (theoretically) doppelgangers of all people on the various Earths, that's not a guarantee. Notice that while there is a Supergirl on Earth-X, there is no Supergirl on Earth 1, and none of the other heroes of Earth 1 on Supergirl's Earth. Her Earth has no speedsters at all.
Also, when they do have parallel people, their lives are not always identical - note the Hunter Zolomon of Earth 1.
If there was no STAR Labs of Earth X, and no accelerator explosion, their Barry Allen, if he even existed, would not have gotten his powers, nor would any of the people who got theirs the same way. Snart didn't get his via the explosion, so he's unaffected.
This would also explain why Ebobard Thawne is the one from Earth 1, and not from Earth-X's future - no Flash, no desire for future-Thawne-X to want to emulate him.