Thanks everyone.
I tried what I could of the above. I can't push anything through because at no point the solder melts all the way through the hole. Tried more flux on the board, flux on the braid, another type of braid (new, but fishy), another type of solder (known good and new). The soldering iron is good. Tried up to 480C.
The best I could get is shallow depressions in both sides of the hole, but the middle remained blocked. I suspect something blocks heat transfer in the middle. Is that a common occurrence, and any idea what and why that might be?
Those drill bits, what range of mm sizes is useful for this?
BTW, in some cases when touching the iron to the hole something creaked. Maybe the board, but not sure. Common thing? A bad sign?