Yes

> Class mobility is a fallacy

This is patently false. Social mobility is entirely possible, although it has gotten harder since the 1980's, but that wasn't your claim.

> that was invented to support the hierarchy

The hierarchy doesn't need "support". Class and social hierarchy has been observed in every culture since time immemorial. The radical and unfounded claim is the notion that we can "remove" it in any meaningful sense.

> Its not an alternative system in any sense.

Moving into a new social class impacts your life in every conceivable way, from the way you speak, dress, behave, to the job you perform. On an individual level, it absolutely would resemble "systemic" changes to both one's material and immaterial life.