How would you picture a version of AoStH/SatAM that was more along the lines of the Mario/Zelda cartoons in adapting the source material? (i.e. visuals, music, SFX) AoStH might have been that to an extent, but I believe its coexistence with SatAM weighed it down some.
Why it wasn't done this way? I don't know, maybe it was because the shows were split and thus some elements like the rings had to be removed, maybe it was because SEGA allowed SEGA of America to write their own lore which thus encouraged DiC to go their own way with their stuff. I think one thing to bear in mind comparing Sonic and Mario shows is that Mario had been around for almost a decade (Donkey Kong came out in 1981, Super Mario Bros in '85), when the Super Mario Brothers Super Show came out (late '89, just before SMB 3 was coming out in America) Mario was already a well-established and his games were familiar in the American pop culture. Sonic came out in 1991 and whilst DiC approached them for the rights to make a cartoon in late '91 it wouldn't be until '93 before work started on both shows. Maybe SEGA weren't comfortable giving out SFX to people, maybe the producers thought they sounded cheesy in the Mario show and didn't want to repeat it, maybe they did but that was just a few pennies out of the budget, they might have just not even considered the idea. Someone's probably asked but I doubt anyone would remember why DiC's Sonic shows never utilised sounds from the games.