Sorry but the Saturn games were not "poor". The AM divisions games were outstanding and offered arcade feel, something virtually no PSX game ever did, across its entire lifetime. The closest to that feel that I ever got was from Psygnosis' Wipe-out, but it still didn't have the refresh. The Tekken series were okay but still didn't feel "arcade" and felt laggy compared to VF2 though the Tekken series was lightyears better than Toshinden, ugh.
None of the PSX 3D fighters had the same responsiveness or arcade feel as the VF series, Fighting Vipers, Last Bronx, etc. and the 2D fighters that were available for both platforms played better on the Saturn. One of the few 3D fighters available for both, Dead or Alive, was better on the Saturn (I had it for both, to counter your Tomb Raider example). The PSX had some clever games but it was a major disappointment and rarely got pulled out at my house, and I had both systems the first day they went on sale.
I played Ridge Racer some and Midnight Club and Rage Racer but they didn't have the arcade feel of Sega Rally Championship. The top PSX track-n-field game was disappointing after playing Sega's high-resolution 60fps game.
Sorry, nope. 3rd party offerings on the Saturn were generally not too good, I'll give you that, unless they were 2D. None of them invested in coding the way the AM divisions at SEGA did. But playing PSX games it was like Sony had never been to an arcade before. NAMCO was likely the most successful but they still seemed like they couldn't quite get there.