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I'm not sure why you think this lattice method does not teach place value. The page you linked to doesn't explain (at all) why it works, sure, but the method is just a clever way of notating place. I would expect any good teacher to get his or her students to understand that, and not just mindlessly fill in numbers.

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I would expect any good teacher to get his or her students to understand that

That's not how I've seen it used. I've seen it used as a procedure for getting the answer. Compare that to the method shown in the npr link, first line: the numerals are 2 and 3, but the numbers multiplied 20 and 30. That's what I mean by teaching place value.

In the the old-school method I learned, we respected place-value by indenting (from the right), optionally writing the zeros.

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