I don't agree. If that logic were true, then hourly payments would be even better.

A weekly payment of $500 would be the equivalent of a monthly payment of  roughly $500*52/12=$2167. If you compare the total interest costs through the life of a mortgage of weekly payments of $500, biweekly payments of $1000 or monthly payments of $2167, it is almost a wash. The difference amounts to a couple weeks' worth of interest. The decision in that case is more aligned with is most convenient from a cash flow perspective. Most folks receive a pay cheque every two weeks, so they may as well pay their mortgage with that frequency.

A mistake folks sometimes make is comparing a $500 weekly payment  to a $2000 monthly payment. The first choice will amount to an extra principal payment of $2000 per year (i.e. the difference between 52*$500 and 12*$2000), which is a big effect as Rob Carrick mentioned. But you could accomplish virtually the same thing with a $2,167 monthly payment.