Methods like get_/set_ do not necessarilly correspond to any actual member variable of an object

Making FETCH and STORE procedures that handle every possible method call would mean

Let me be more explicit. If you change something from a simple accessor that you reach through the tie methods like $foo{'bar'} = 7 to a method call, you will have to either change that calling code to something like tied($foo)->set_bar(7) or else put a special case in the FETCH method so that it calls "set_bar()" if the key is "bar". This would break the abstraction in the former case or compicate the code in the latter.

UPDATE: as requested, an example, using your code. Say we change the definitions of get/set_foo in MyThing to something like this:

sub set_foo {
  my ($self, $newfoo) = @_;
  $self->[1] = localtime(); # track last change
  $self->[0] = $newfoo;
  $self->_recalculate_bar();
}

sub get_foo {
  $self->[2] = localtime(); # track last access
  return $_[0]->[0];
}
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After this change, the $clunker code using MyThing requires no changes at all. On the other hand, the tied HashAPI version, must change the accessing code from this:

$slick->{foo} = 'Some value';
print $slick->{foo}, "\n";
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to this:

$slick->set_foo('Some value');
print $slick->get_foo(), "\n";
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You could avoid needing this change by changing the FETCH and STORE, like this:

sub STORE {
  my ($self, $key, $val) = @_;
  # assuming this class has more than just 'foo'...
  unless (grep { $_ eq $key } @CLASS_KEYS ) {
    croak (ref $self) . "has no $key member";
  }

  if ($key eq 'foo') {
    $self->[1] = localtime(); # track last change
    $self->[0] = $val;
    $self->_recalculate_bar();
  } else {
    # set normal keys somehow
  }

}

sub FETCH {
  my ($self, $key, $val) = @_;
  # assuming this class has more than just 'foo'...
  unless (grep { $_ eq $key } @CLASS_KEYS ) {
    croak (ref $self) . "has no $key member";
  }

  if ($key eq 'foo') {
    $self->[2] = localtime(); # track last access
    return $self->[0];
  } else {
    # get normal keys somehow
  }
}
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That would quickly become cumbersome though.