One day, over ten years ago, I purchased a Conn C soprano sax. My intent was to cover an oboe part in a local community band. Shortly after I purchased this instrument, the band found an oboe player.
Sarge, the World Wide Saxes founder and tech, said it was the hardest instrument he’d ever brought back to life. For example, he said the springs were like rusty cheetos! But he did an amazing job.
When I got the instrument back it was like new. I had the original mouthpiece and purchased a couple of new c soprano mouthpieces, I really wanted this instrument to work. But the inherent intonation was so bad. I sadly realized that it would take a lot of time by a better musician than myself to use this instrument in an ensemble. So I sold this instrument at a loss and moved on.
The Bis Key Chronicles 
C Sopranos are much better in theory than they are in the concrete, although….that might be the best place for most of them
Is there anything inherently wrong with it that makes the intonation bad? I heard people on YOUtube play them with good intonation. I’m thinking about buying one but if I do I want it to play, I have enough collector’s items. I know the smaller the instrument, the more difference pushing or pulling out the mouthpeice makes. If you have to pull or or push in too much it will throw the entire turning off.
The average Joe or Josephine would not be able to make this instrument sound good. I often suspect a pro can make a 2X4 sound magical. YMMV.
I have a soprano that was flat. If I pushed the mouthpiece in enough to get it in tune the so called throat tones would be off. I finally did the string trick and it really works. I’m getting ready to take it in for a repair I’m going to talk to the tech to see if a more permanent solution can be found.
Interesting what you say about your experiences. I finally managed to get one. It too is a Conn, and yes its intonation can be a bit squirrely.
So far I have all of about 6 or so hours on it since its restoration, so obviously I don’t have the horn anywhere near figured out. That said, as primarily a large-horn player I find the trick is not to “over blow” it, and keep a really tight control control of my facial muscles. Doing those 2 things I have all but 4 or 5 notes 100% in-tune. Those that are out, are about 20-30 cents #. I am slowly trying to build my endurance up on it. So far I can do about an hour before my embouchure is finished.
The embouchure control it requires reminds me a bit of your Soprillo Jim. Combined with the back-pressure it produces, these little beasts are wickedly hard on the head to play. But the results are worth it. The tone is lovely, and quite different from a Bb soprano–and least from mine. 😉