I am just an amateur and a hack at that, but my professional field is biomechanics, and I get a little into ergonomics. The thumb is just not designed to take pressure from the side. The rubber pad will distribute the pressure around the thumb a bit and if it relieves your pain, that’s brilliant.
This post is for reference if it does not work and for others seeking relief. The thumb is really strong through its length but weak from the side. I’ve known guys that could do push ups on their thumbs and one could do one-arm push ups on his thumb! I also know a thumb lock from martial arts that only requires perhaps 5 to 7 lbs. of pressure on the side of the thumb to bring someone to their knees.
My first sax teacher was a retired woodwind professor. I picked up my alto and lifted it up to my mouth leaving the strap a little slack. He said, “No, no, no, let the strap carry the weight. Gently push the sax away from you with the pad of your thumb.” When I got a soprano, however I basically held it like a clarinet, which had required that I get a “Thumb Saver" to help with the thumb pain from bearing weight on the side of my thumb during my brief tenure playing clarinet. I too experienced a lot of thumb pain on soprano.
I started watching some of my soprano exemplars, especially in the way they brought the soprano to their mouth. Two used neck straps, and in slow-mo you can clearly see that their thumb pushes out pivoting on the neck strap as the mouthpiece comes toward the mouth. After getting a Brancher Crescent neck strap, which puts the weight on the stable thoracic rather than the unstable cervical vertebrae, I was very happy with this position. My only comment is that the neck strap length has to be perfect, and so a lot of people will not like this method.
Since I became used to this position, however I’ve found, without the strap, if I simply hold the soprano on the pad of my thumb as I would with the strap and rotate my wrist to let the mouthpiece come to my mouth, this is also a comfortable position that I can maintain for a long time. I still think of the thumb as pushing the bottom of the sax away from me rather than lifting the sax by the hook with the side of my thumb. This change in thought pattern alone will change your mechanics enough that it should make a difference in your comfort.