Your question hits a critical historical assumption that's actually more subtle than it first appears. The Michelson-Morley experiment assumed an "aether wind" because 19th-century physicists believed the aether was a universal, stationary reference frame pervading all of space — not something that could be "dragged along" by matter.

The key assumption was that aether was fixed and absolute, like a cosmic ocean through which Earth moved. If matter constrained or dragged aether locally (your intuition), then yes, you'd measure light relative to the local aether frame, not the universal one — and the experiment would've shown null results for the wrong reason.

But physicists explicitly rejected "aether drag" theories after earlier experiments (like Fizeau's water experiments) suggested partial dragging at best. They needed aether to be stationary to explain stellar aberration and other phenomena.