If you are near Southern California, I would recommend taking it to Mike Yamasaki, who can give you good advice on the blade and the polishing process. What I can tell you today is that the blade comes with an appraisal paper (origami) from Hon'ami Kō'on, dated 1662. Kō'on says the blade comes from the Sa lineage (he doesn't specify the exact smith). The blade was already shortened and had lost its signature by this time. Then, in 1953 (maybe January, can't quite tell), Japanese sword scholar Kashima Tsutomu reaffirmed the sword was of the Sa lineage via his own appraisal. Then, in June of that same year, the NBTHK also appraised the sword as coming from the Sa lineage, and gave it a "Precious" designation (one of the designations that predates the Tokubetsu Hozon designation).
The current "Sa Hiroyuki" appraisal comes from the latest NBTHK appraisal, dated 2021, in which it received the Tokubetsu Hozon designation. So, no doubt the sword is of the Sa school, but there may not be absolute consensus among the Japanese sword world as to the Hiroyuki appraisal. This may be one reason the sword currently does not have a Jūyō designation.
My gut feeling is that if the sword were a super strong candidate for Jūyō, the sword dealer Seiyūdō would have submitted it themselves in order to take advantage of the potential profit that they could have received by selling it as a Jūyō blade. But I am not an expert, and that "gut feeling" is sort of a lazy person's conclusion.