In particular, any code that runs fine with strict, will continue to run fine, and in exactly the same way, if you remove the line use strict;.

Most of the time, yes. But not always.

use strict 'refs'; has a run-time effect. There can be programs that rely on symbolic references throwing an error. A program can rely on such behaviour.

use 5.010;

sub ok {
    # remove the "use strict;" and try again...
    use strict;
    local $@;
    eval { $_[0]->[0]; 1 };
}

say ok('foo');
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It is unlikely that this code appears like this in production somewhere, but it can very well be hidden somewhere less obvious.