Why are all changelings Shapeshifter creature type?
You're right in that even without the Shapeshifter subtype, they would have that subtype in any relevant situation, due to Changeling being a characteristic-defining ability.

That said, there are a number of Shapeshifter creatures that don't have the Changeling ability, and many have existed before the ability was introduced in Lorwyn as a carbon copy of Mistform Ultimus's ability (a card that's curiously not a Shapeshifter).

Changeling is not an evergreen mechanic, meaning it is only used in several sets - as of now, Lorwyn and Modern Horizons. The subtype Shapeshifter however has been used frequently throughout sets and indicates that it is more than a mechanic to the designers - it is a part of the lore of the multiverse, and despite one might more easily interpret function into "Shapeshifter" than "Kor", it is nevertheless a race of beings within the world.

This is hinted at in the phrasing of a DailyMtG artcle by Devin Low:


  [...], the designers had given the five-card cycle one more thing to share besides having all creature types. "How would a race of Shapeshifters fight?" the designers asked.