Is that what you mean by Sonic R "looks terrible" edge? Technically the game is easily in the top ten most impressive 3D engines for Saturn, but it dissappointed most Sonic fans obviously. Sonic Xtreme could have been great I think, it preserved the spirit of the 2D gameplay while playing to the Saturn's graphical strengths in an impressive way.
Geoffrey, I've never disagreed with you about the dithered transparency issue in those terms, of course there's nothing preferable about it. It's only that I understood why it is the way it is, and made my peace with the fact early on. Tomb Raider is the perfect example, and I've used it enough times before, the Saturn game is the only version I would ever want to play, given a choice. The coherent image, both in the (absence of) texture warping and the geometry precision (no seams) puts the PSX to shame for me; nothing spoils suspension of disbelief like erratic image quality. Beyond that is the more nuanced and atmospheric lighting, particularly with the water effects - also the water distortion effect lacking in EVERY other version - and as you mentioned not just a better draw distance but also a more consistent frame rate compared to PSX - which drops into single digits very often in the big open room areas. Some of the colors are nicer on PSX though, and the minor translucency effects are of course better, and the textures are definitely higher res overall, but for me there is still no contest. TR also makes an interesting sort of negative example of the most powerful distinction of the Saturn, because it is effectively powered only by the VDP1, but still holds its own with the PSX.
A couple great examples of that potential discrepancy are Mass Destruction and Robo-Pit, which run at twice the frame rate on the Saturn vs the PSX versions (only single player for Robo-Pit) all because of their reliance on large detailed bitmaps for the terrain, which is exactly what the VDP2 excels in. Essentially that one "Mode 7" processor is effortlessly doing the same work that, in the other case is consuming half of the PSX's fill rate for it to approximate. I had some grasp of these tradeoffs from the start of my Saturn relationship - since VF2 also shows that power to fine advantage - and so I accepted them.
Just a little more about Powerslave/Exhumed and Lobotomy. That's also a case of a game that, even though it doesn't seem to capitalize on the VDP2 power it still more than holds its own with the PSX example. Extrapolating from the hints I have, and the fact that I don't see how that engine could be using the same trick as Burning Rangers for the blending seen in the flare lighting effect; I think there is some priority protocol with the VDP1/sprite alpha blending, which may still be connected to the VDP2 circuitry, like it interrupts the sprite data fetch to pass through the blend function maybe. Between Ezra Dreisbach being a code monkey par excellance, and the more easily sorted composition of the image, it may be a special case where the "transparency bug" can be properly accounted for and the blending itself still incurs no performance overhead. Hard to explain especially when I don't have a firm grasp of the details, but I can roughly make sense of it in theory.
So to wrap up this train of thought, I haven't ever looked into it all that deeply, but my understanding of the basics from well before the Saturn days helped me to understand the references I saw as they accumulated into a pretty solid picture. I have had the thought of scrounging through SEGA-16 for more details on the graphics spec but too lazy / not in the mood so far. Heh
So one final response, in Sonic R the soft draw-in (fogging) effect is indeed true blending and it is accomplished by the exact same function as the translucency effect of the special track - which is why it can't do both - and also every other (true) sprite translucency effect you see in games like Guardian Heroes and Silhouette Mirage. EDIT: also I remembered a couple examples of true sprite blending seen in PDS, the underwater connecting tubes in Uru, which IIRC appear to operate the same as this current example; but also in some cutscenes like the dogfight with Atolm after Azel ambushes Edge at Uru, you see the nice energy barrier effect. - And it is basically a no-cost effect, because all the video data is collected for final output regardless, if the alpha-channel data is non-zero for any given pixel the display processor simply sends the sum value of all lower priority channels (in this case the background bitmap) and the sprite color value through the blending function. Because there is no read-modify-write overhead it is in one respect much more efficient than the alpha-blending the PSX or any other modern GPU employs - so when it is usable it is a cheap effect for Saturn, while blending is always costly for a standard architecture and therefore why you typically see flat color effects on the PSX - but it has inherent limitations. Which is why all these overall tradeoffs are excusable, in the context of the technology of the day. I'm still guessing that the Saturn has two alpha channels / blending circuits, in some form. But maybe not, and in either case the dithered transparencies are understandably unavoidable for some situations, given the practical technology of the day. EDIT: well that and the fact that the Saturn is a kludge architecture informed more by SEGA's Super Scaler arcade legacy, which was always about throwing more processors on a board for purpose built functions I think!