V⁶/₄ is used instead of I⁶/₄ for a few reasons.
First, it was the original way I learned it, from the Harmony and Voice Leading book I used in college. But I believe it has another benefit, in that it tell us the actual function.
V⁶/₄ is known as the cadential ⁶/₄ and occurs at cadences. In C major, if we look at the notes, we have G in the bass, then C, then E. What makes this different from a I ⁶/₄, is where the notes go, and why they go there.
With a cadential ⁶/₄, we are saying that this chord, is not just it’s own chord, but belongs to the next chord, which is most often a V⁵/₃ or V⁷ chord. In the V⁶/₄, the E will move down to D, and the C will move down to B.
With a I⁶/₄, you most often see it as a passing chord, where the bass note, G, is moving as well.
It is really because of the tendency of the cadential ⁶/₄ to move to a standard V chord, that we say it is an embellishment of the V chord. Really it is one big harmonic function – cadential.