Bearing edge plays a big part. The tone is fine, I’m just looking for something different
I totally get what you're saying. My first kit, a PDP FS Series birch kit in fusion sizes, had SUPER ringy toms. I attributed that ring to the really sharp bearing edges, which from what I could see were a single 45 degree angle. At the time Phattie Drums, a small boutique company that I believe is now defunct, was making drums with precision bearing edges - dual 45 degree edges with what they called the "Phlat Spot" in the middle. Using that as an idea and a guide, I put together a sanding table with a big piece of MDF and two different grades of sandpaper, and I sanded down the sharp edges on those toms. This is a diagram I made clear back then and posted to the Pearl drummer's forum about what I'd done.
I didn't want the sharpness on the outside edge, so once I'd flattened them down, I took a wood block and some fine grit sandpaper and gave it a very light roundover. I know that some people might be horrified by the idea of personally altering a bearing edge in that way, but you have to figure, they were only a $400ish shell pack, and I didn't figure that I was going to ruin them. If all else failed, I could always take them to Keith Larsen (Baltimore/Maryland Drum) and have him recut the bearing edges. (Keith was literally the next door neighbor to my brother-in-law, but didn't realize just who he was until after I started playing drums, and after he'd moved - I actually knew him before I knew he was a drum builder.)
The end result, at least to my ears, was pretty remarkable. The 12" tom, which had been horribly hard to tune, would tune right up, and the drums themselves lost that overly ringy/boingy sound. It smoothed out the sound - it was warmer and seemed fuller. I'd kind of wanted to keep them, but I made a deal with my wife that I'd sell them to help fund the purchase of a new kit, the Pearl SMX Session I still have and play.
In any case, I understand the part about the role the bearing edge plays, and I hope you find what you're looking for.