My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF.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due to Joseph Ayoub (in a more general axiomatic context), and was also worked out independently by Oliver Röndigs (unpublished but available online); Voevodsky stated the axiomatic result that Ayoub proves in an unpublished note, but as far as I know there is no record of a proof by him.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, 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.