this is an interesting study of the use of vibrato in classical sax -
https://etd.ohiolink.edu/rws_etd/document/get/ucin1384428227/inline
it seems like Marcel Mule is to blame for the 'wow and flutter' school of classical vibrato
"At the time of the saxophone’s invention in the mid-nineteenth century, it was uncommon for wind instrumentalists to use vibrato in their playing. Even as vibrato became more accepted in vocal art music, many still considered it inappropriate and in poor taste for winds. The earliest saxophone studies and method books, including publications by Kastner (1844), Cokken (1846), Hartmann (1846), L. Mayeur (1867), Klosé (1877), and A. Mayeur (1896) made no mention of vibrato. Carl Weber, in his 1897 method described the ideal saxophone sound as “unwavering.” Even in the early part of the twentieth century vibrato was still not associated with the saxophone. In 1917 Ben Vereecken published his Foundation to Saxophone Playing, one of the most important pedagogical studies of its time. In the section titled “What to Practice,” he stated: “Avoid the ‘tremolo’ or ‘vibrato’ style of playing. See that your tone is absolutely clear and pure.” Later he wrote, “Avoid letting the tone tremble by the loosening of the lips.” As late as 1920, George Koehler’s book instructed to “avoid any semblance to tremolo or vibrato.”
In Europe there is one person who undoubtedly influenced the development of concert saxophone vibrato more than any other. This of course was Marcel Mule. Mule began playing saxophone at age seven, but also seriously studied violin in his early years. He later claimed that his studies on the violin taught him a great deal about tonal
homogeneity, virtuosity, and articulation, concepts he helped bring to the concert saxophone world. One may assume that Mule developed his saxophone vibrato style from the violin, an instrument that was already developing a reputation for extensive vibrato use during the time of his childhood. But according to Mule himself, this was not the case. Mule claimed that his vibrato all started with jazz.
Even though Mule had studied the saxophone since c.1908, he did not hear jazz played on the instrument until the early 1920s. When he first heard saxophonists play this new type of music he found the tone remarkably strange, even calling the rapid vibrato many early jazz players utilized “dreadful.” Anyone familiar with the vibrato style of French-Creole saxophonist Sidney Bechet, widely regarded as the most influential jazz saxophonist of the 1920s, can probably understand Mule’s bewilderment. Nevertheless, Mule was inspired to develop his own vibrato.
In 1923 Mule began performing with jazz groups around Paris for extra income. He described himself as a decent jazz player, but not a great one because he did not “favor the choruses.” Later on he admitted that jazz taught him about the saxophone’s power and versatility. Eventually Mule was able to succeed in finding the undulating vibrato characteristic of the local jazz musicians. Though Mule had learned this new technique, he still played classical music without vibrato, treating the instrument much like a clarinet. It was in 1928 that everything changed. Mule had been performing with the Opéra Comique in Paris for a short time when they were scheduled to play a ballet
titled Evolution, by Edouard L’Enfant. L’Enfant was a pianist-composer who dabbled in jazz and liked to include pieces such as foxtrots and blues in his compositions. There happened to be a saxophone solo in the ballet with the composer’s instruction “very vibrant.” After being unsatisfied with Mule’s interpretation of the solo on the first playthrough during rehearsal, L’Enfant asked Mule to play the solo like he had heard him play jazz. Mule feared using his jazz vibrato would lead to a “scandal,” but obliged the composer anyway. Much to his surprise, the performance received exceptionally high praise from his peers, some who even suggested he always play concert music in that manner.
From that point forward, Mule used vibrato in all of his playing. It is apparent that he considered the day he initially used vibrato with the Opéra Comique as somewhat of a turning point in his career."