If I remember correctly, the Diamond card's BIOS didn't support VBE 2.0 out of the box, but the SciTech universal VBE driver that was bundled with the 11th Hour detected the card and handled the VBE 2.0 support.
I later learned that universal VBE drivers shipped with some games tended to be outdated and cause problems with some newer graphics cards. The trick in such cases was to manually load a more up-to-date version of the universal VBE 2.0 driver before running the game. My guess is that upgrading the video BIOS to a version with integrated VBE 2.0 would also work, if such an upgrade were available for your particular graphics card. But this still wouldn't work for card that didn't support linear frame buffers, which was a hardware feature.
TL;DR: you need a PCI or VLB card with linear frame buffer support to run The 11th Hour, and if the card doesn't have VBE 2.0 (or higher) integrated in its video BIOS and the universal VBE driver bundled with the game doesn't recognize the card either, you'll also need to load a VBE 2.0 or 3.0 driver that supports the card, before you run the game.
Just try the last version of SciTech Display Doctor or UNIVESA that was ever released. If that doesn't support your card, I'm not sure what will.