The Liberal Republicans did not have an issue set that had staying power. They were for ending Reconstructing early (which got Greeley the Democrat nomination) and they were mainly anti-corruption. All parties are pure as the driven snow in opposition, but once in power, it is hard to maintain, human nature being what it is.
Basically they were abolitionist democrats in the post-Civil War era. They were never going to last long.
And I agree that three party system is tricky in the US. The only way for it to work would be for a highly regional base of support that could consistently elected 7 to 9 Senators who had the balance of power, and bunch of Representatives.
And the Presidency can be split, sort of. Imagine a coalition situation sort of like what exists in Australia. A major party candidate cuts a deal with the third party, who supplies the VP spot on the ticket. Could work.