[microtherion]

Yes, the advantage of gridding seems to be that it decomposes the multiplication into simpler steps.

I would suspect, however, that if you habitually had to do long multiplication, the old method would be faster (precisely because you do a bit more in your head and less on paper). Multiplying two 5 digit numbers involves adding up 5 numbers instead of 25.

But what has changed over the last 40 years is that NOBODY does habitual long multiplication anymore, because calculators are ubiquitous. Therefore, the technique is purely a building step for further understanding, and picking an algorithm optimized for casual users of multiplication is appropriate.