Can a Magsafe i/o Board charge a battery externally?

The magsafe will not put out the 16.5v - 18.5volts it supplies until the SMC and the magsafe are "shaking hands" and the SMC tells the Magsafe it can. Your lack of the LED(s) lighting on the magsafe indicate that the SMC and power adapter don't know the other exists. To answer the question in your title, not touching the rest yet, is not with the intended power supply.

If you have replaced your left I/O board with a good one. Then either that board isn't talking with the logic board or the motherboard has problems. With the information given, I would start by checking the magsafe connector for residue or debris. The center contact (the small one) on the magsafe is where it communicates with the SMC. After that I would double check the connections of the power cable that connects between the I/O board the logic board. If those are good I would inspect the cable ends and the socket on the logic board for any bent or burnt pins with a jewelers loupe. I would also inspect the area around that socket and the area on the other side of the board directly underneath it for any signs of burning such as discoloration or charring - Inspect both the board and components in those areas. This is a 12 layer board with most of the power running through sub levels.

Please give us the last 3 characters of your serial number so we can determine your exact machine and better be able to assist you. Please give us any history of the machine passed on to you. Has this ever experienced liquid damage?