The battery is not essential to boot any original compact. How badly corroded is the battery enclosure? Serious leakage may have damaged the adjacent analogue board and power supply.

Most of the previous discussions here have addressed "wonky but working" analogue boards and power supplies, and I recommend that you take a look at previous threads about faulty video, intermittent restarts etc. They won't help you to diagnose the problem, but will give you an indication of the type/scale of repair that will be required.

Anyway, welcome to 68KMLA, and good luck.