... how come $ref->[$_] can’t be written as $ref[$_]?

Is it because $ref points to a reference of an array .... Whereas $ref[2] would be the third element of the array @ref ...?

Exactly. In Perl, @foo and $foo (and likewise %foo and etc.) are distinct variables which may have the same identifier, i.e., name. (Update: They are distinguished by their $ @ % sigils.) E.g.:

Win8 Strawberry 5.8.9.5 (32)  Mon 11/16/2020 14:26:26
C:\@Work\Perl\monks
>perl -Mstrict -Mwarnings -l
use Data::Dump qw(dd);

my $foo = [ 99, 42, 137, ];
dd '$foo', $foo;

my @foo = (qw(foo bar baz));
dd '@foo', \@foo;

my %foo = (qw(cero zero  uno one  dos two));
dd '%foo', \%foo;

print $foo->[2];
print $foo[2];
print $foo{'dos'};

^Z
("\$foo", [99, 42, 137])

("\@foo", ["foo", "bar", "baz"])

("%foo", { cero => "zero", dos => "two", uno => "one" })

137
baz
two
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(BTW: $ref does not point to a reference to an array, it is a reference to an array. :)