I check for continuity between GND and VCC, I get a tone, indicating a short. The meter shows it at around 40 ohms resistance.
Is this really a short (resistance is not 0 ohm)?
The tone doesn't indicate a short, it indicates a resistance less than a certain value, often about 60 Ohms. The exact value varies across different makes/models of meters.
A low DC resistance value between 5V and ground is normal, that's not the problem.
It's common with A2000 battery corrosion damage to have poor/intermittent contacts with the DIP68 68000 socket and/or DIP40 ROM socket. If the socket contacts look even slightly corroded (i.e. not shiny), then replace the sockets. The IC leads usually need cleaning with a fibreglass brush and isopropyl alcohol if they're not clean and shiny. I'd not recommend attempting to clean damaged sockets, as any such effort is often temporary, and you're likely to experience intermittent problems long term.
The white screen can indicate all sorts of different things, such as one of the CIA hardware timers not operating. Which can be caused by a loss of the 15kHz or 50Hz timing signal to the CIAs, or a failure of one of the CIA ICs. Though instead of guessing, DiagROM is a useful tool to easily understand that particular failure. But sort out bad IC socket first as that can potentially get you going.
