Because no ROM SIMM was ever made for the IIci, the on-board ROM must be selected with a jumper at W1. However, without it the IIci behaves like a kerbstone. You write that you are getting raster and mouse tracking, which are both well down the road after successful POST, although still in the 'hands' of Start Manager in ROM.
In such circumstances as you describe, no chime indicates bad audio—not necessarily the speaker—and it reads as if the caps in the output or the vicinity thereof have messily yielded up the spirit. Other large smoothing capacitors under the PSU may well have done likewise. But it is perfectly possible that a defunct PRAM battery is advising the system that the sound-level setting is zero. Given a valid startup volume the next signposts on the way should have been Happy Mac (found valid boot blocks on a startup volume), splash screen (the blocks pointed to a valid System Folder) and Desktop (the System and Finder got it all together).
Your immediate need may be a complete and thorough board wash, but if removing the goo clears the misbehaviour, there is no alternative to total replacement of the electrolytic caps to avoid recurrence of trouble. Then only will you be in an assured position to consider such nuances as booting.
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