The problem with removing that early sequence in LOTR is that the Hobbits don't get the old knives from the Barrow they use as swords. Tolkien stresses that, had Meriadocs' sword not been a product of the Numenorean craft of Arthedain 'wound about with spells for the bane of Mordor', he could never have wounded the Witch-King severely enough for Eowyn to finish him. Nor would Peregrins' blade been able to gut a Troll at one stroke at the Black Gate. This substantially reduces both Eowyns heroism and the menace of the Nazgul.
The Sinbad/Arabian Nights subgenre has the advantage that its base stories are less well-known in the West than Greek or Norse legends. This gives the film-makers more scope and freedom to write their own tales and populate them with monsters etc.