|'Beren and Luthien in the Court of Thingol and Melian.' copyright 2017 Donato Giancola|
'I will tell you the tale of Tinúviel,' said Strider, 'in brief – for it is a long tale of which the end is not known; and there are none now, except Elrond, that remember it aright as it was told of old. It is a fair tale, though it is sad, as are all the tales of Middle-earth, and yet it may lift up your hearts.'(FR 1.xi.191)
Aragorn's words here indicate that the Tale of Beren and Lúthien was not remembered and told 'aright' in places other than Rivendell. Given the multiple, unfinished or abbreviated versions of the Tale Tolkien wrote, he may well be poking fun at himself here.
To say that the end of the Tale is 'not known' is not the same as to say that the lay is unfinished. Indeed the hobbits later hear it sung 'in full' at Rivendell (FR 2.iii.277). What Strider says here, at Weathertop, shows that he fully understands what Sam grasps only later on, that they're 'in the same tale still. It's going on', because the 'great tales never end' (TT 4.viii.712).