I think you've nailed it by considering the carry-overs. With the new method there's less opportunity for error. The old way, there are two sets of carry digit

I think you've nailed it by considering the carry-overs. With the new method there's less opportunity for error. The old way, there are two sets of carry digits: those arising in the initial multiplying, then more in the final adding up (though not in the example given). Sometimes I would try to remember them; sometimes I would write them down in various places; frequently I forgot or muddled them up somehow. Another benefit is: the new way gives you a sense of the magnitude of the final answer more quickly. Approximations are useful. Contrast with the old method which begins by multiplying the least significant digits.