indeed as wise monks above (choroba and poj) already said the (sub)tree is not parsed (never realized this fact btw) in the moment you catch <name>name1</name> if instead you put your logic on the site element it works as you expect, just doing $_->delete if $_->children ('name[string() =~ /name2/]');

use strict;
use warnings;
use XML::Twig;

my $xml = q(
<sites>
<site siteid="ONE">
   <name>name1</name>
   <address>address1</address>
   <contact>contact1</contact>
</site>
<site siteid="TWO">
   <name>name2</name>
   <address>address2</address>
   <contact>contact2</contact>
</site>
</sites>
);

my %handlers = ( site => sub { 
                                # my ($twig, $cnt) = @_;   
                                # $cnt is unuseful: iirc $_ is mapped 
+$_[1]

                $_->delete if $_->children ('name[string() =~ /name2/]
+');

} );          
my $twig= new XML::Twig( PrettyPrint   => 'indented', twig_handlers =>
+ \%handlers);

$twig->parse($xml);
$twig->print;

# output:

<sites>
  <site siteid="ONE">
    <name>name1</name>
    <address>address1</address>
    <contact>contact1</contact>
  </site>
</sites>
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