"if you overload it [a solid-state relay] it may short."

Or it may blow open circuit.
But anyway, by definition an "overload" is any load beyond the ratings of the device and is, therefore, likely to damage the device and stop it working!

"why on earth would you overload it!"

Through failure to read & understand the ratings of the switch and/or know the nature of the load

"You must have some sort of protection in your circuit for overload."

Obviously not - he's already admitted blowing up relays!

In general, a mechanical switch is more rugged than a semiconductor.
A semiconductor junction/bond wire has a very low heat capacity ("thermal inertia") - usually much lower than any fuse.
Therefore semiconductors often make very good fuse protectors! ;-)