A short on the secondary side will happily draw maybe 5A from the a.c. supply and the fuse will never blow.
Just try it and see.
A colleague of mine was once convinced that he could protect a toroidal transformer of around 30VA with a 1A fuse on the primary. I warned him but he insisted. We put a short on the secondary and we rapidly had one transformer with nicely melted insulation after a few secs ! Didn't wait to see the final outcome.
Nope - nothing is allowed to smoke and catch fire.
Graham