Does True Seeing reveal an Elder Oblex’s simulacrum’s true form as an ooze?

I’m currently playing in a campaign where we’ve encountered an Elder Oblex. Before we ventured into its lair I twinned True Seeing onto myself and another party member.

Later on it was revealed that a pretty suspicious humanoid male we met down there was actually a simulacrum created by the Elder Oblex’s Sulfurous Impersonation ability.

The description of the ability is:

As a bonus action, the oblex can extrude a piece of itself that assumes the appearance of one Medium or smaller creature whose memories it has stolen. This simulacrum appears, feels, and sounds exactly like the creature it impersonates, though it smells faintly of sulfur. The oblex can impersonate 2d6 + 1 different creatures, each one tethered to its body by a strand of slime that can extend up to 120 feet away. For all practical purposes, the simulacrum is the oblex, meaning the oblex occupies its space and the simulacrum’s space simultaneously. The slimy tether is immune to damage, but it is severed if there is no opening at least 1 inch wide between the oblex’s main body and the simulacrum. The simulacrum disappears if the tether is severed.

From the description of Truesight in the Basic Rules/Player’s Handbook (emphasis mine).

A monster with truesight can, out to a specific range, see in normal and magical darkness, see invisible creatures and objects, automatically detect visual illusions and succeed on saving throws against them, and perceive the original form of a shapechanger or a creature that is transformed by magic.

So, while not necessarily the traditional definition of a shapechanger (typically associated with polymorph’d creatures or lycanthropes), clearly the Elder Oblex is shapechanging beyond its natural composition, and should be detectable via True Seeing, and the characters who had True Seeing active should have seen an ooze in the guise of a man.

Is this a reasonable deduction or a stretch? Perhaps there is there a more specific definition of what is or isn’t a shapechanger somewhere that I am missing?