I'm sorry for the stupid question, but I'm really a newbie with amigas... how can I make the test program you linked run on the amiga?
Unfortunately replacing the C17B will prove almost impossible, since the pads basically exploded with the capacitor and there is nothing to solder to there. Moreover I don't have any ceramic capacitor lying around. The most I can do would be installing an electrolytic capacitor in its place (by this I mean simply between the first and the last pin of U17), but I don't really think if it worth the trouble (and risk).
Update: I recapped the machine, no noticable difference (even the audio is still very faint). Most capacitors were not really bad. I noticed a few traces under C303 (in particular D15) that were almost black. They tested ok with the multimeter, but I might consider using wire there. The same for a trace just under C334 (near the rca jacks for audio)
It seems to me that your board has been shorted out, probably on the metal shield. I wonder if someone had trod on it when it was on or something hence pushing the traces through the mylar and onto the shield. If not trod on, then the keyboard heavily banged or similar.
To use that program, you'll need a program called lha first in order to uncompress other files from aminet (similar to zip files):
http://aminet.net/package/util/arc/lhaThis is a self extracting archive. It should create an executable called 'lha' which you should copy to your C directory, or in your case just to the Ram Disk (or on a floppy as you have no hard disk presumably).
Then, with your lha archive to hand (memcheckbh.lha on a floppy for example) and a copy of lha in either RAM (Ram Disk) or C, double click on the memcheckbh.lha icon and when the 'execute a command' requester opens, insert 'lha x ' before the archive, so that the text box will read 'lha x memcheckbh.lha' without the quotes and press return. Lha will then extract the archive into the same path as the archive itself (floppy in your case).
[generic] 250 740 33.8% -lh5- 38b1 Jan 29 1995 MemCheck.info
[generic] 1675 2984 56.1% -lh5- 465a Oct 15 1994 MemCheck/MemCheck
[generic] 1833 3700 49.5% -lh5- 99e3 Oct 15 1994 MemCheck/MemCheck.doc
[generic] 310 643 48.2% -lh5- 01cf Dec 8 1994 MemCheck/MemCheck.doc.info
[generic] 494 884 55.9% -lh5- 8c23 Dec 7 1994 MemCheck/Product-Info
Five files should extract as above (you'll only see four though on the Workbench screen as one is a hidden '.info' file (an icon image). To read the documentation you can boot your Workbench disk, look in the Utilities drawer and run the program called More. Then when the file requester pops up, point it to the text file on your floppy DF0:MemCheck.doc.
The docs should tell you how to run the program and what options (if any) are required.
Just double clicking MemCheck and pressing OK or return should at least give you the options to run the program, or with a bit of luck it will run without options and start a memory test.
To do any of this you will need some way of getting the files from aminet to your Amiga. If you don't have a PC with a floppy drive and a copy of CrossDOS or fat95 on the Amiga side, you'll need a Serial or Parallel cable to transfer files (normally serial). There are kits you can buy with cable and software included. Alternatively, did your A600 come with any software at all? Normally they do come with some Magazine coverdisks or pd software and they will have lha already on them in the C directory. Also, if you're lucky you may find a memory check program too!