I find his vibrato irritating, too. But if you listen to other musicians from this time, especially vocalists, I think you'll find that he was not that far out of step with the practices of his time.
Sidney's vibrato stems directly from the New Orleans Créole clarinet school which produced him. That school goes back to the 19th century, the famous French Opera House in New Orleans, supported by the well-heeled New Orleans Créoles who would send their children to France to be educated.
Your comparison with vocalists is pretty apt, but again, it's primarily a French influence. Early French operatic recordings from before WWI show this vibrato; it is still to be heard in more modern French popular music, such as with vocalists like Piaf and Aznavour.
Sidney took the Créole clarinet vibrato and applied it to soprano sax; since it is much easier to produce vibrato on a sax than on a clarinet, he developed a very wide vibrato.
It's often been suggested that he went over to soprano from clarinet because he wanted to
lead groups in performance and be able to put noisy trumpeters in their place. This is just (im)pure speculation by self-styled jazz "critics" who, most of them, haven't the first clue about actually
playing an instrument.
(There was a story commonly told about Sidney and Wild Bill Davison standing literally shoulder to shoulder at the microphone at a recording session in the 1940s, both wanting to run the show, and neither willing to take a back seat for a moment. This is the sort of claptrap beloved of jazz "journalists." It's utter bulltish, or so Wild Bill said when I asked him about it during a gig he played at my local pub back in the late 1970s.)
It has also been suggested that he used the broad vibrato as a mechanism to control his pitch on an instrument notoriously difficult to play in tune, which may have some element of truth in it, though it's far from the full story, as it takes no account of the influence of the New Orleans school of clarinet.
Whatever the reason, with his volume, his cutting tone and his vibrato Sidney dominated ensembles he played with, as can be heard on record.
Ultimately, it's up to the listener to decide whether he/she likes Sidney or not, just as it's up to the same listener to accept or reject such notables as Hodges, Bostic, Parker, Coltrane, and Gorelick.
As for the method of producing such a vibrato:
It helps if you've learnt to sing, with proper diaphragmattic breathing. It also helps if you've spent a few years on clarinet learning to produce a loud volume to compete with a triple forté trumpeter standing beside you, thereby developing strong lungs and diaphragm.
To play, you hold the instrument up, out from your body at at least a 45-degree angle; you drop your lower jaw and you use diaphragmattic breath control allied with a certain amount of jaw feathering.
It takes months, even years to get this right - in doing it, you go against all "good practice" and teach yourself a mountain of bad habits which are extremely difficult to unlearn later on, as I found out to my cost…
Why do it? Sidney was his own man with his own sound. We can admire him as one of the greatest saxophonists of all time, but that doesn't mean we should try and sound like his clone, any more than we should try and sound like Gorelick.
Far better to get your own sound, be your own self, and play what
you need to play, to express yourself and nobody else, presuming you're not just an empty shell, that is.