The Glenn crew wore the Disco uniforms so that would fit into your theory.
But the captain of the captain of the USS Gagarin was wearing the Disco uniform, also. Then again the crew aboard the Cabot were wearing the colorful uniforms. Its interesting from an in-world explanation perspective, though I am sure, in general, the costuming department is happy to be shut of the complicated 1st and 2nd season uniforms.
edit: add USS De Milo as a ship wearing the Disco uniforms.
The Glenn, Gagarin and De Milo being under the purview of Section 31 makes sense, and those with the DIS uniforms but no black badges are unwitting resources for Section 31 to use to serve the Federation. For whatever reason, the Federation saw it as acceptable to give Section 31 free reign to do whatever ops or experiments they wanted.
It would also help to explain the line of no one seeing a Klingon in 100 years when the Federation and the Klingons are supposed to be engaged in unremitting hostilities for 30 years by the start of DIS. It would mean that is really Section 31 that has had limited communication with the Klingons in 100 years, while the Federation has continually had to deal with the hostilities in the meantime.