Mow it, leave the clippings.
"The soil is hard and compacted" - Mow it and leave the clippings. Put the organics back, and this will help. If weeds do sprout from this, they're shooting roots in and helping the hard and compacted soil, helping the compaction. And you'll mow them again and both feed more organics and weaken them. Weeds are green, and your soil wants green.
You might be able to play with the height some, depending on if/what grass exists, and if you want to keep it. "Generally speaking" I'd still mow to a height that isn't scalping (so you have some green to look at) until the fall when lawn renovation can take main stage (assuming cool season grasses).
While you're figuring things out, simplify and feed the soil: Mow, leaving the clippings. Don't make more work for yourself with bagging. (Unless you really want to, and then I'd suggest packaging it up and sending it all to me for my compost pile!