Even if Yoda had won...

[Rocketboy]

Even if Yoda had won and beaten Sidious, would it have been too late for the Jedi and the Republic anyway?
By the time Yoda and Sidious face off, Palpatine had already gone before the Senate, told the tale of the Jedi's "assassination attempt" (with the physical injuries to boot), swayed the Senate's opinion heavily against the Jedi, and turned the Republic into the Empire.
If Yoda had won, it seems that despite destroying the Sith (because without Palps, Vader would have slow-roasted on the coals) it may have appeared as if the few Jedi still alive after Order 66 were back to finish the assassination they failed to do earlier, which could have made Palpatine a martyr for the new Empire.
And even if the Jedi were able to explain all of Palps shenanagins, I doubt the Senate would trust the Jedi enough to believe it.

So, was it a "damned if we do, damned if we don't" scenario?

[JimJamBonds]

In a sense that is correct that it was a no win situation for the Jedi. With Palps gone that would have gone a long for the Jedi to return to the good graces of the Senate etc. Although you still have the universe in a state of war so all sorts of problems could have come up. I'm sure the remaining Jedi and members of the Senate that were loyal to the order would have done what they could to show everybody what Palps really was and what he did.

[Kidhuman]

Mind trick the lot of them, it would work.

[Slicker]

I kinda thought the same thing before. It makes sense since Palps told of the "treachary" of the Jedi and what not that if Yoday killed Palps it would have made it harder if not impossible for the few remaining Jedi to survive any length of time.