Reddit actually had a few good points against it: 1) Flying Nazguls, first and most obvious - and Sauron would know they are coming, being unoccupied with war with Gondor. 2) By skipping Rohan and successively Gondor, Aragorn never becomes a king, kingdom of humans still most likely falls. 3) Nobody solves the Saruman threat, so in the meantime he becomes more and more powerful - probably crafts his own ring or in an…
Furthermore, I believe it is only included in the extended edition of The Two Towers (but I could be wrong on that point).