Many Amiga games did run natively from the hard disk but it usually depended on the kind of game, the developer, the game's original platform if converted and number of floppies. RPGs, adventures, simulations, wargames and strategy games had a good chance of running from the hard disk but more action-based games such as shoot-'em-ups, platformers, beat-'em-ups and arcade racers were less likely.
Hard disk installable games usually came with either custom install programs or scripts that ran through the Amiga's own installer application.
Arcade-style Amiga games often hit the hardware directly without going through the Amiga's operating system and had their own file system and format, so the the files can't be viewed in Workbench and appear as NDOS (non DOS) disks. WHDLoad came later and solved this problem by using disk images and accessing the custom files through the disk image file as well as providing a common system.
Therefore you could have installed The Secret of Monkey Island 2 (11 floppies), Indiana Jones And The Fate Of Atlantis (11 floppies), Formula One Grand Prix, Civilization, Colonization and Ultima VI: The False Prophet on your Amiga's hard disk using their own installers although they were also brought into the WHDLoad system as well for extra features such as the quit-to-Workbench key.