Doesn't the current to run the trains heat the wires?

I hope not. Miles of cabling putting out enough heat to melt ice might be less than entirely efficient.

Separate de-icing systems are available, but I'm not sure how widely they're used.

Shouldn't there be a small generator onboard for emergencies, to at least keep the toilets running? Perhaps a tank of natgas for heat?

Trains have very limited space, and weight is an issue.

In the UK failed electrics are hauled to safety by troubleshooter diesels. But if a long stretch fails, the diesels have to do multiple runs to rescue multiple trains. Which takes a while.

There were plans to create a hybrid diesel/electric next generation Intercity train in the UK, but weight and space requirements made it impractical solution.

I expect when nuclear fusion becomes small enough to fit into an engine car these problems will be solved. But until then catenaries, with all of their faults, remain the best choice.