Bet the grass grows back quickly when you have moved them
As for the leaves in the OP's garden I would say it doesn't really matter how you get them into the soil, they will still do some good when you do. Whether you mulch with them as they are, mulch with them partially composted, or dig them in they will end up as leaf mould in the soil; that means better consistency, moisture retention, and drainage. There is an experiment pouring excess water through various mediums, leaf mould will finish draining first, but hold on to more of the water than the others.
What leaf mould does not do is supply lots of nutrients, there are some, of course, but not lots.