You've commented out an essential statement :)

        $_[0]=$handle
            unless defined $_[0];  # pass up to caller
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This sets the $fh in open my $fh, ...  Without it, $fh won't be the handle created in *CORE::GLOBAL::open.

Putting this near the end of that routine should suffice, but if you want the overridden open() to work for other forms too (not only 3-args open with lexical file handle), you should also re-enable the qualify_to_ref(...), i.e. the routine should then look like

    *CORE::GLOBAL::open=sub (*;$@) {

        use Symbol();
        my $handle;
        if (defined $_[0]) {
            $handle = Symbol::qualify_to_ref($_[0], scalar caller);
        }
        
        if (@_ == 1) {
            CORE::open $handle
                or warn $! and return 0;
             }
        elsif (@_ == 2) {
            CORE::open $handle, $_[1]
                or warn $! and return 0;
             }
        elsif (@_ == 3) {
            if (defined $_[2]) {
                CORE::open $handle, $_[1], $_[2]
                    or warn $! and return 0;
                 }
            else {
                CORE::open $handle, $_[1], undef  # special case
                    or warn $! and return 0;
                 };
             }
        else {
            CORE::open $handle, $_[1], $_[2], @_[3..$#_]
                or warn $! and return 0;
             };

        $_[0]=$handle
            unless defined $_[0];  # pass up to caller

        tie *$handle,"TieInput",*$handle;
    };
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With this change, I get the following output:

$ ./953722.pl
opening in1.txt:
TieInput:  $_s = \'in1 A
';
$s = \'in1 A
';
  at ./953722.pl line 107, <$In> line 1.
TieInput:  @_a = (
        'in1 B
',
        'in1 C
'
      );
@a = (
       'in1 B
',
       'in1 C
'
     );
  at ./953722.pl line 109, <$In> line 3.
opening in1.txt and in2.txt:
TieInput:  $_s = \'in1 A
';
$s = \'in1 A
';
  at ./953722.pl line 118, <$In> line 1.
TieInput:  $_s = \'in2 A
';
$s = \'in2 A
';
  at ./953722.pl line 120, <$In> line 1.
TieInput:  @_a = (
        'in1 B
',
        'in1 C
'
      );
@a = (
       'in1 B
',
       'in1 C
'
     );
  at ./953722.pl line 122, <$In> line 3.
TieInput:  @_a = (
        'in2 B
',
        'in2 C
'
      );
@a = (
       'in2 B
',
       'in2 C
'
     );
  at ./953722.pl line 124, <$In> line 3.
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