Pulling out my 3-ring binder of old Apple TIL articles, this document was #21011:
"Power Macintosh Computers: Some Don't Support Mac OS 7.6."
The recently-introduced Power Macintosh computers ship with either System 7.5, version 7.5.3 (Power Macintosh 4400/200) or System 7.5, version 7.5.5 (Power Macintosh 5500, 6500, 7300, 8600, and 9600 series computers).
None of these systems supports Mac OS 7.6, and Mac OS 7.6 will NOT install on Power Macintosh 4400, 5500, 6500, 7300, 8600, or 9600 computers.
A coupon to receive the OS 7.6.1 full-installer CD for $9.95 s/h was included in the box with those computers. These CDs contain an OS 7.6.1 System folder to boot computers not supported by 7.6, and have 13 sub-installer folders within the "Software Installers" folder. You should be able to determine which 7.6.1 CD you have, by exploring its contents. When launched, the installer first checks the computer's gestalt ID, to determine if it's compatible with the software. That's why the OS 7.6 CD installer stops dead in its tracks. As a workaround, you could try booting from the OS 7.5.5 hard drive and run a small program (there were two available on the web - check VersionTracker for "GestaltLab") that temporarily changes the machine's ID. You choose/substitute the gestalt ID # of a supported computer to fool the installer. If successful, you'd be able to install OS 7.6 from your CD and then run the 7.6.1 Update from the other CD. The gestalt-modifying utility is a temporary alteration and the computer reverts to its actual ID # after restart. For this reason, you'll likely need to run it again, prior to running the 7.6.1 Update.