If you specify a column in your INSERT statement, it seems at least SQLite (but I guess SQL) won't use the default. I've added a third column, which is not mentioned in the INSERT statement and it always picks up the default value:

#!/usr/bin/perl
use strict;
use warnings;
$|=1;  #turn STDIO buffering off

use Data::Dump qw(pp);
use DBI; 

my $dbh = DBI->connect("dbi:SQLite:dbname=:memory:","","",{RaiseError 
+=> 1})
    or die "Couldn't connect to database: " . DBI->errstr;
      
$dbh->do ("CREATE TABLE ScoreCard
             (  id        INTEGER PRIMARY KEY AUTOINCREMENT,
                Score     INTEGER DEFAULT 11,
                My_Info   VARCHAR DEFAULT '<none>'
             );
          ");

my $insert_row = $dbh->prepare("INSERT INTO ScoreCard(Score)          
+                     
                                VALUES(?)");
                                
# desired goal: get Score in the DB to be 11 if its not specified.

$insert_row->execute();        #row 1  Score=undef
$insert_row->execute(undef);   #row 2  Score=undef
$insert_row->execute('');      #row 3  Score=null string
$insert_row->execute(333);     #row 4  Score=333

my $get_all_rows = $dbh->prepare("SELECT * FROM ScoreCard");
$get_all_rows->execute();

my $aref = $get_all_rows->fetchall_arrayref;
print "all rows="; pp $aref;

__END__
all rows=[
  [1, undef, "<none>"],
  [2, undef, "<none>"],
  [3, "", "<none>"],
  [4, 333, "<none>"],
]
[download]