My thoughts against non-manufacturer rebuilds / retips have softened somewhat over time - of course partially out of necessity, with more old masters passing on (more reports to come in the next several years, I’m sure). At some point I’ll need to have a Koetsu rebuilt by 3rd party (had a few rebuilds done by Koetsu) - and that’s OK; life goes on. I’ve seen a couple 3rd party rebuilders using what looks like the exact same Japanese-sourced cantilever and stylus assembly used by Koetsu, and these should be a strong consideration (Ana Mighty Sound in France, Sửa Kim Than Cartridges in Vietnam). That said, I probably wouldn’t mind a Fritz Gyger on boron either. Groovetickler in USA also seems to do great work. Also check out "Delta667" on the vinylengine forums. He is a true fanatic in exploration of cartridge design & rebuilding! There is so much to learn from the massive "Cartridge Close-Ups - Show Your Shots!" thread, and people there even get along!
Once you replace original parts with something different, you go down the road of making a hybrid, and the spectrum of "good, but different". So what parts matter? Ortofon’s Kontrapunkt and Cadenza series (Kontra being the predecessor of Candezna) offer an interesting exploration into this question - each line had 4 models with basically the same motor generator and body (with some minor differences), but featuring different combinations of cantilever and stylus. Given my experience with these models, plus experience with different Koetsus, I’d place the magnet type and then cantilever material as having the highest sonic impact. Next is body type. With various Koetsu stones being very similar - that differentiation is more subtle than outright changing body material types (like from wood to stone or metal). Coil wire and stylus - I’d rate these as more subtle changes to the sound.
Of course the coil (windings and armature), damper, and yoke arrangement probably have largest impact on sound, but that is probably harder to isolate for a controlled comparison. I had a vintage Koetsu Onyx (non-platinum) rebuilt by Koetsu, and it sounded quite different - both to how it sounded before, and next to a modern Onyx Platinum. The older coils sounded more lush and "romantic".