I think you're asking why I think my method is better than the long division you mention. I've used that method too, in the past. Hmm... On paper, they seem pretty similar, but I think it is easier to answer the question "What multiple of 7 can we add or subtract from the final digit n to get 0?" (a memory lookup if you know single-digit multiplication and pay attention to final digits) than "What's the biggest multiple of 7 we can subtract from the leading digits mn?" (requires a couple of tries and comparisons).
If the first two digits happen to be "21" or "42", etc., a multiple of 7 by themselves, then I will probably notice and subtract that out. But that's fairly rare (1/7 assuming uniform distribution).
Overall, I find working from the small end more rewarding. Note that I never have to add or subtract anything larger than 7 x 5, and, 7/10 of the time, I don't have to add/subtract anything that isn't 7, 14, or 21. (I sometimes choose to add/subtract larger numbers to avoid carrying a hundreds digit.)
And note that I do this with 13 and 17 and so on as well. Then long division becomes even harder, whereas it's no harder to determine which multiple to add/subtract (as you just need to look at the last digits; 13 and 23 are like 3, and 17 is like 7), and maybe just a bit harder to calculate this multiple (e.g. last digit = 5 -> add 23 x 5 = 115; last digit = 9 -> subtract 23 x 3 = 69).
If the first two digits happen to be "21" or "42", etc., a multiple of 7 by themselves, then I will probably notice and subtract that out. But that's fairly rare (1/7 assuming uniform distribution).
Overall, I find working from the small end more rewarding. Note that I never have to add or subtract anything larger than 7 x 5, and, 7/10 of the time, I don't have to add/subtract anything that isn't 7, 14, or 21. (I sometimes choose to add/subtract larger numbers to avoid carrying a hundreds digit.)
And note that I do this with 13 and 17 and so on as well. Then long division becomes even harder, whereas it's no harder to determine which multiple to add/subtract (as you just need to look at the last digits; 13 and 23 are like 3, and 17 is like 7), and maybe just a bit harder to calculate this multiple (e.g. last digit = 5 -> add 23 x 5 = 115; last digit = 9 -> subtract 23 x 3 = 69).