Perfectly valid concern! ;-). I actually very much enjoy our discussion.
Let me add a few points then to leverage your understanding of the advantages involved.
Yet another great advantage of OO-ish exception handling is be that different types of errors could be dealt with at different 'layers' of your code. For example, I might have a system exception in my code handled by Error::Sys. Such an error could be thrown but the inner-most subroutine and picked up and processed only a few levels up the call tree. (yet not at the very top of it... say at the main:: package). There may be a number of other exception types/categories that are only handled by certain layers of your code. In such case, OO-ish way is slightly batter compared to eval/die.

Making same with eval/die would (if possible) be somewhat cumbersome (at least the way I see it :). In order to weed certain die messages at various layers of my code, I might have to institute a mechanism to make sure that every die message has a particular keyword in it. For example, for a system error I might have "SYSTEM: error text". Then, later in the code (or up the call tree), I'll check for a match with that key word. And if matched, I'll process it accordingly instead of leaving it to be handled by upper-layer code (sorry if my terms sound amusing :-> ..). With OO, you may simply do something like this:

# called from sub foobar() which in turn
# is called from foo() which in turn...
sub do_something {
... 
try {
    stuff();
} catch Error::Sys with {
        # handly any System error right here
};
# any other error will go up the call tree..

}
[download]

With eval/die similar code would have to be implemented as follows:

# called from sub foobar() which in turn
# is called from foo() which in turn...
sub do_something {
... 
eval {
    stuff();
};
my $err = $@;
if ($err =~ m/SYSTEM:/) {
        # handle any System error right here

} elsif ($err) {
    # any other error will go up the call tree..
    die $err; # rethrow
}

}
[download]

I'm sure there's tons of other examples that could be brought up. I hope that the ones I mentioned are reasonable ;-)