Since I'm now only reading from $PROG_STDERR when there's data to be read (!eof), reading nothing does indicate an error, right?

It it was a file... But we're talking about a pipe. It might not be closed when you check eof, but it could close after you started trying to read from it.

Or so I thought.

eof apparently does a read to find out if more data is coming.

$ perl -wle'
    open(my $fh, "-|", sleep => 5) or die $!;
    print eof($fh)?1:0;
    print defined(scalar(<$fh>))?1:0;
    print eof($fh)?1:0
'
<<eof blocks for 5 seconds here>>
1
0
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This behaviour is very different than the behaviour of stdio's feof, but it's documented: eof "Returns 1 if the next read on FILEHANDLE will return end of file, or if FILEHANDLE is not open."

So, you're right. Reading nothing in that situation would indicate a reading error (and thus, you can print out $!).

(No consideration was given to non-blocking handles.)