Why do you want to improve Tomb Raider? Even on PlayStation it has both improvements and reductions to get that final result. Why even assume 30fps? Shenmue clearly runs way, way worse in the tech demos (but it would have been perfectly fine for the things it'd offer instead of framerate had it ever released back then, OoT style) and hell, so does Tomb Raider in many spots. Idk why you're choosing snapshots of the very first area like that's the whole game, in either quality or performance anyway, even the training room shows more complexity (but lol @ using emus to judge anything about real hardware), never mind areas like
St. Francis' Folly (this is a rough PAL version video regarding a save bug but it speeds past some nice views,
playthroughs I found of the US version are like speedruns and beeline all the jumps etc. without any looking around) that include both way, way more complex geometry and may have several enemies lurking and roaming and other interactive and moving elements on top.
That's great and all, but it still doesn't detract from my point where the rushed release, small team and unoptimized argument is used, but that is only valid if it gives you something back that's extra, in this case if it is something that could equal or surpass the PS1 version, which in this case, we take performance out of the equation so Who cares if what I took screenshots from was running at 15fps or 120fps on the emu, the only way to get something extra is to improve the visuals.
On the Saturn we either need to increase texture quality by non-interpolated (x4 geometry) tessellation, increase geometry detail by tessellation, or both, and as I previously explained the multipliers for each are beyond beta to release optimization IMHO, which is what would be required to get the Shenmue Saturn demo running on stock Saturn hardware.
In your Shenmue shot there's nothing even impressive there, it's a simple area with like one equally simple extra detail element making it more than a plain square (the corner pillars or whatever, everything else is painted on). You're just assuming Ryo's backside has a crazy amount of polygons but it's not like that model would be as seen in the in-engine cut scenes with detailed hand, facial and eye expressions etc. and I doubt that would be real time lighting from the window to the walls, just textures, I don't think any game that era has that detailed lighting/shadows to actually take the shape of the window with the gaps and everything and if any do they're definitely the exception, I don't think anyone expected that to be the case. Hand placed varied textures to fake it in otherwise way smaller scope and scale indoor areas like that (hell even nowadays we have games where indoors have better dynamic in this case lighting than outdoor scenarios) certainly seems doable, if painstaking as everything manual is vs modern automated systems.
I didn't say it was impressive - PS1 could easily have achieved that - but the requirements for Saturn to achieve that with texturing being such a performance hit compared to PS1, the texture quality in that demo are above Tomb Raider, and as I've already set my stall out, it would need 4x the geometry detail non-interpolated to get that increase, despite having more taxing geometry on display, and not because I think there is vastly more geometry in that playable Shenmue screenshot, but because Ryo's on screen size and texturing means there is more overdraw by his larger on screen projection and there is more geometry in his model because of the detailed texturing on this model by comparison to Lara.
And the lighting looks per vertex dynamic, which again allegedly hits performance, and even if not and baked reinforces my assertion that the texturing is at a level beyond the Saturn without a addon accelerator board.
But okay, back to how some early title, a studio's 2nd game on it, one of the very first in this full 3D open roaming style granting it and its series a higher than legendary status (on the platform it reached later, lol) for video games, maxed one of the according to this thread most difficult to develop for systems, because you analysed an emulator shot and spec sheet (yet these geniuses somehow didn't max the easy to develop for PlayStation with their same-era games which are super close on both systems yet still did way more with sequels and other games on that eventually, it's magic)
You massively underestimate the ingenuity of development at the time (particularly British development where they were ahead of the curve) back then and the push towards more 3D experiences since games like the Skull on 48K Spectrum, Argonaut Software doing Starglider on ST/Amiga(eventually doing the mode 7 SuperFx chip on SNES) or Elite, 4d Sports Series, Hard Drivin,etc and the game on Saturn and PS1 is consistent with what they released on PC as I recall at the time, pretty sure it was prior to console release, so both version IIRC were ports of a finished PC game that ran on PC CPU with 2D accelerators, so hardly the rush job of an unfinished game debuting early on a launch platform.