Am I correct in my thinking?
How do you distinguish between a fuse that has gone open improperly and one that has blown? I don't think you can conclude that the fuse "failed" without replacing it and seeing how much current the board draws. It is possible that the fuse is marginally rated and just blew after a long period of operation at or near its rating, or that there is or was an overload or short.
In any case, I believe a ball of solder in the middle of a glass fuse is not a blown fuse in the normal sense. I think perhaps the fuse gradually failed as it heated up, while it was failing. This leads me to suspect that nothing on the board caused the fuse to fail.
If nothing on the board caused the fuse to blow, it's reasonable to assume that the board might be OK.
Nevertheless, when my replacement fuse arrives, I'll be able to solder the replacement in place and test the board. Then I'll know if I was right.