It would make $bar-XX>method() safe if $bar is undefined, but what if $bar is a scalar? Or an unblessed reference? Or an array ref? or a coderef?

The poll question specifically states it's for attempting to call methods on undef. I'm fairly certain that it's not intended to gloss over attempts to call methods on arbitrary scalars and especially not on references of the incorrect type. Those should definitely remain errors.

Would the same operator work for fetching elements of array refs and or keys of hashrefs, or only for methods?

Interesting thought... It would make sense to me, at least, that if an arrayref/hashref is undef, then all of its elements are implicitly undef as well.