A maintenance visit is a slot on a route. It is sized for a lawn already at height, beds already at their lines and shrubs already in shape, and the work is to hold all three exactly where they are. A property that has been left for two months in July is not that job, and no amount of hurrying makes it that job.
A cleanup takes longer, uses equipment a mowing visit does not carry, and produces a volume of material that has to be dealt with. It also cannot always be finished in one pass, because a lawn that has to come down in stages needs a second cut a week or two later before it is back at a height a schedule can hold.
So the honest sequence is a cleanup first and a schedule after it. Putting an overgrown property straight onto a weekly route means the first several visits are spent catching up, the finish is never what anybody wanted, and the lawn takes the damage in the meantime.