However, after the loop, there doesn't seem to be a way for the program to detect that a file wasn't processed. In real life, that could be awkward.

Simple solution: use strictures.

perl -Mstrictures -e'while(<>){1;}continue{print "$ARGV\n" if eof;}; p
+rint $!' file1 file2 file3
file1
Can't open file2: No such file or directory at -e line 1, <> line 1.
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That should make it pretty obvious even to the most obtuse of users that something has gone badly wrong.

If you are after a less abrupt, more hand-holding approach then yes, I think it probably best to roll your own.