I’ve actually commented on this in a similar thread once before. There is a genre of classical saxophone that most people aren’t aware of. I have used a saxophone quartet at church. They accompanied hymns, the mass parts and played preludes, fugues, postludes etc… The tone color of a saxophone quartet is (to many people) remarkably similar to a pipe organ. As such, I had quartets play Bach, Gabrielli, Palestrina, Tallis etc… The saxophone quartet, with the exception of the string quartet, is the most uniform sounding common chamber group that exists. It is also extremely well suited for playing with or completely replacing an organ when necessary. I have never used jazz saxophone in church though.
If there is/was a ban on wind instruments, it seems somewhat peculiar as the pipe organ in nothing more than a giant wind instrument. Its mechanism of creating a tone is largely identical to any other wind instrument, just with a better set of lungs.