... the screen displays garbage. During the turn on Lisa one beep appears (sometimes two short beeps).
I have a second good I/O board and after inserting it into the same Lisa there are no problems. So the issue is definitely somewhere in I/O board.
Since it works with a different I/O Board, I agree the CPU and RAM are (probably) ok.
Since the board used to work, progressing corrosion damage may be a likely suspect, so a visual examination of the area where the battery used to be, down towards the card edge may reveal something.
IIRC, one beep means the CPU didn't find any RAM, which, if caused by the I/O board, suggests a bus contention issue. I imagine there are quite a few possible causes, so another Lisa with similar symptoms might have different causes.
A clue is that the CPU can control the I/O board enough to generate a beep, which suggests the address and data bus are at least partially working for writing data.
The video circuitry reads the screen directly from RAM, so in cases where the 68k CPU is not running but the address and data bus are working, the screen can have a steady but random image. In your case, is the screen image changing/moving, or is it steady with one random image?
If it is changing, then the I/O board would appear to be interfering with reading RAM in general, but if the screen is static, then the I/O board may be interfering only with 68k access to RAM.
It is useful to know you have an oscilloscope:
As a first suggestion, I would set the scope to trigger on the falling edge of /AS (Pin 3 of U8A) with a time scale that permits viewing through to the following rising edge, then probe the signals on the I/O board that connect to the CPU board, looking for signs of contention (signals that are between 0.8 V and 2.0 V for more than a moment) while /AS is low, or any that are permanently stuck high or low.
Signals to check:
Data lines
Address lines
UDS
LDS
DTACK
VPA
READ
INTIOHopefully a further clue will appear, or perhaps someone else has some more specific advice?
HTH & good luck!