It's really hard to tell from afar where a leak may be!
FWIW, yes, the boiler might last a lot longer, but then again, it could be shot. Depends somwhat on how it was setup and run. Also, it's likely that a boiler that old is radically oversized. AN oversized boiler is usually very inefficient. The local utility company probably has a rebate program to replace it with something much more efficient. Whatever you do, if you go this route, do NOT just replace it with something the same size, as something maybe 1/3 the size or maybe even less would be enough, and save both a lot on the installation cost (bigger boilers cost more), and in efficiency, and in comfort. Idealy, a boiler runs constantly with just enough output to keep you warm as the start/stop cycles tend to waste lots of energy and create a lot more wear and tear on the parts.
Do you have an autofill valve? Is it open? Is the pressure stable (it probably should be if the autofill is open, but that masks leaks)? Is this a tank with an embedded water heater coil? If so, it could be potable water that is leaking, not boiler water.