Well in a way that makes sense - the fuse is an extra, and at somewhere around a 12 amp surge when it starts up that is a weak link.
I have old
sun system ballast from the 90's, and it doesn't have a fuse anywhere inside or outside. I have replaced the ballast transformer and capacitor and added an extra square foot of heat sinking to it; so I happen to be familiar with it's innards to know there is no protection fuse inside.
Mine is magnetic, as yours is; it's possible there is no danger in eliminating the fuse and twisting those two wires together with a wire nut. I would watch it like a hawk while testing it though. If it heated up once - it might have bad windings or another problem like a bad capactitor or igniter.
Replacing any internal parts of a magnetic ballast requires some skill to be safe - there are large currents present on the inter connecting wiring that will heat up and cause problems if not joined properly.
Another problem with replacement of ballast transformers is mounting alignment.
The Universal kit I got for mine did not align, and I tried adding a few spacers between brackets to get the holes right to bolt it in; that caused terrible humming and vibration.
For a minute I thought my new ballast kit was bad; but once I drilled some extra holes in my case to accept the new ballast transformer all of that ceased. So a very tight fit at the mount point, along with solid electrical connections, is critical to success in repairing your own magnetic ballast.
Hope this helps.