So. Couldn't replicated it not working with the battery out in the shop, because everything just turned on fine there for some reason. Though there were clear issues, that were gone if the battery was out.
Along with other things the battery is doing, even though it isn't doing classing things like swelling, it seems to be the problem and is likely sending some feedback that is screwing with other things too, like it has partly damage the power brick on my new cord from the look of it, already.
It's working just fine right now with the battery out, but it may not continue to. And there's no way to tell until it may break again, if it is more damaged.
So basically... I'm back already but I might not stay that way. I'm in limbo. It's great fun.