Ok the system works fine with the external speakers, but it still refuses to sound the chimes of death. Could really caps be the cause for no chimes of death?

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The Macintosh Classic, nor ANY 68000 Mac plays the Chimes of Death. Your Classic is functioning normally except for the bad cap that is preventing the internal amplifier from working.

You need to open it up and clean the logic board, preferably replacing the bad caps, before the leaking cap goo causes more problems.