Do check the traces around the battery with a multimeter... however, that's not definitely the problem.
Take out all the SIMMs except for the 2MB one towards the centre of the board - this is the Chip RAM which is necessary to run at all. Now (with the power off, obviously) put one 4MB SIMM into the socket next to your Chip RAM. Boot up the machine and check how much memory you have. If it's 4MB Fast RAM, put it in the "Working RAM" pile, otherwise if it's zero, put it in the "Suspect RAM". Repeat with the other three.
If
any of them give 4MB Fast RAM, your board is good, and you have a bad SIMM. if
none of them give any Fast RAM, your board is bad.
All is not lost, however... it's probably just a matter of finding the track that's not connecting any more and sticking a wire on it to make the connection again.
Even then, if you can't fix it, there's still hope! Your board is working with its Chip RAM, and that's most important. All you need to do is to buy an accelerator with local RAM (pretty much any accelerator does) and bingo! you have a full complement of RAM again (probably

).
It's not over yet, anyway, so don't despair! You've done the most important thing (removing the battery) and the machine is basically working, so it's just feeling a bit dodgy, not dead!