I see your point Jim, but one other factor...if all of my accounts were weekly I wouldn't be able to get to them all at times. I would have to drop some accounts, hire someone from outside the family or add a mowing day to my schedule, none of which I want to do at this point.
And keep in mind that mowing is only a small piece of the pie on some of these accounts...the cleanups, plowing and other maintenance and extras account for a larger portion of my income on them. I billed one of them over $3,500 last year...I left that one out of my figures because it's billed together with a small $30 monthly (their cottage "grass" parking area). Mowing only accounted for about 25% of my billing on that account last year. The guy isn't cheap, he just doesn't fertilize or irrigate, so it doesn't need cutting often.
The bottom line is that my bi-weekly accounts fit in well with my operation and I don't want to replace them with weekly accounts...strange as that may seem. It all depends on your situation. The barometer for whether or not an account is worth holding onto for us smaller operators is different than for someone like you. For me it's largely how I feel when I pull up to the account to service it and how much it brings in per year. If I dread pulling up to the account and/or it's less than $1,000 per year it's a candidate for the axe.