Swapped all the socketed chips in my 3000T with the ones in my 3000 desktop (known working) except for the memory chips without any change. sigh....
The 3000T is quite robust and i have worked on quite a few in my time. things i have found that do go wrong:
Power supplies(start here..check it under load. +5v rail should be at least 4.92v or they wont boot in my experience.
12v should be at least 11.92 or the same can result. you're mileage may vary here if its borderline.
check the oscillators are seated well,they are socketed.
Battery damage can be the cause of the black screen and the caps light quits toggling after about 6 tries. Look for broken traces to the rom,cpu,etc in the damaged area.the cpu is not likey to be dead,just a trace somewhere. If you have a A3640 accelerator,you might be able to change jumpers and test with that to get the machine up.
Bad roms: they have failed(bit rot)? or are making bad contact/corrosion from the battery leak. if you see green corrosion,don't try and clean the sockets etc,replace them.
Bad chip ram.. sometimes can cause a black screen. Seems some Toshiba brand zips go bad in the chip ram,i have replaced ram on 5 machines with this problem in the past. change ram as a absolute last resort
Denise socket, i had one intermittent machine with a bad Denise socket,it seems it had a pin broken between the socket and the board that make contact sometimes, you could not see it and it looked perfect.. this was a fun one to find! It was likely a factory defect and gave the original owner hell.
From what you describe,you have a broken trace near the leak. they can be a pain to fix.replace any sockets near the area and check every single one in the area for continuity!
use proper plcc pullers and not screwdrivers..messing up plcc sockets will cause more grief than anything.
sadly the battery has been known to cause problems in amiga since the mid 90's.. its a shame people didnt take them out then!.
good luck.
