Why does the team not know who the Reverse-Flash is?

No one could possibly recognise Eobard Thawne even though if they knew him.

Jefferson, Nate, Amaya have absolutely no scope to even know who Eobard Thawne is, let alone recognising him.

Sara was dead (killed by a drugged Thea) when Barry and his team was facing the menace of Reverse-Flash.

Ray Palmer teamed up with Barry to stop Brie Larvan; he had nothing to do with Reverse-Flash then. He died (?) in an explosion while mending his A.T.O.M. suit before Reverse-Flash showed up in his real identity.

Mick Rory came as a new partner of Snart in the second season; so he is out of the discussion.

The question all boils down to whether Martin Stein knew of the identity of the speedster.

He was present and replaced the position of Wells in STAR Labs when Wells was absent and was later apprehended for being the Reverse-Flash.

He also knew Reverse-Flash is Eobard Thawne for he was present when Barry disclosed his identity.

But, did he see Thawne's real face?

When Eddie shot himself, Thawne returned to his original face before being erased and vanished.

It could be seen in the desktop present in the lab where Stein was inspecting and watching the wormhole.

However, Thawne appeared in his real face for a very short time before getting disappeared. And also there was sort of haste going on as Eddie shot himself and possibly Stein wasn't too attentive to notice his real face; moreover the camera was positioned angularly so that even though in the desktop, the vanishing of the Reverse-Flash could be viewed, it could not be distinct enough to give a clear view of Thawne's real face, thus averting Stein to know his real identity.

In the second season, Stein already departed along with Jax before the time-remnant of Thawne again appeared in the Central City.

So, in a nutshell, we can conclude that, no one of the present Legends ever got the chance to either know about or see Thawne's real face.

As for why they didn't ask Barry during the crossover, it's possibly because there was already a major threat of the Dominators and already they were tensed after hearing their lives getting altered or impacted by Barry's mistake of creating the Flashpoint.

Also, as Phantom42 in his comment asserted, even Barry couldn't have guessed who the speedster was as there are simply many that appeared so far (and would appear) and also not crossing out the possibility of a speedster from any of the Earths of the multiverse.