Why are the first two alternatives not capturing quoted test strings?

Because your position is still at the string start so you can't use a lookbehind (or if you do it will never match). Simpler just to use this:

use strict;
use warnings;

use Test::More tests => 3;

my @strings = (
    'Now is the time',
    '"Now is the time"',
    "'Now is the time'",
);

my $regex = qr/^['"]?(.*?)['"]?$/;

for my $input (@strings) {
    my ($match) = ($input =~ /$regex/);
    is $match, 'Now is the time', "$input matched";
}
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