The primary reason in my mind is that you want to have multiple procedures with which you can double-check work later. Knowing only a single procedure makes for very fragile understanding; on the day that you make an error, you have no way of recognizing or fixing that error. So I would claim that you want at least two or three methods of sanity-checking an answer. (E.g., a rough estimation, an algebraic solution, and a graphing solution.)
Secondly, in many places you would use the initial, tedious procedure as motivation and appreciation for how much more powerful and elegant the later procedure is. Likewise if students have difficulty understanding the sophisticated procedure, then $X$ can be used to confirm that $Y$ does give correct results.
Thirdly, if you have a disagreement with someone over a result, you should have a more basic shared understanding that you can fall back on to similarly check that the answer is correct another way.
That said, in my community-college classes I find that students frequently do resist this, and either zone out on the secondary procedure, or occasionally become actively hostile. Actually, just this week I updated my statistics final (which I've been using for a few years) to require graphing checks of all the problems. Previously that was only recommended, not required -- the weaker students skipped it, basically just hoping they were right, and had no way of knowing how far off they were in most cases. I'm hoping if they know that the secondary check will be required, they will then be forced to develop that skill, have better intuition, and produce better results.