My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result was not proved by Ayoub nor was it known to Voevodsky, and it is different from the result of Drew–Gallauer (which is not about 6FFs). The existence of the 6FF is of course due to Joseph Ayoub (in a more general axiomatic context), and was also worked out independently by Oliver Röndigs (unpublished but available online); Voevodsky stated the axiomatic result that Ayoub proves in an unpublished note, but as far as I know there is no record of a proof by him.

**ETA:** As Peter Scholze explains in the [comments](https://mathoverflow.net/questions/463730/who-proved-the-motivic-6-functor-formalism#comment1204257_463754), he was in fact referring to Drew and Gallauer during the talk (who prove a universal property for motivic spectra as a lax symmetric monoidal functor on schemes, not on correspondences). In particular, I think there is no claim that motivic spectra satisfy the universal property in Ola Sande's question if "6-functor formalism" is interpreted as a lax symmetric monoidal functor on correspondences.