When the mulch is damp, it will break down faster than compared to a rotten tree branch. Watering mulched gardens on a regular basis will ensure a dry state ideal for breaking down the mulch. Cut everything that grows in the yard from now until the time the lawn is planted, either with a blade weeder (usually, the typical rope is not strong enough) or a lawn mower or trim with a sharp clipper or alternatively spray with herbicide. Goats kill trees by surrounding them, and their navigational habits lead them to eat more woody plants and small, leafy trees.
If you have a grassy area where woody plants were removed, the tenacious perennial vegetation has likely formed a dense lawn that will naturally be inhospitable to new plantings and should be carefully monitored before seeds enter, either with burning or with full investment tillage. It would be the same for a vegetation cover as for the lawn, except that with the grass, you cover with hay over the top (or you can get a removable planting cloth) and with vegetable covers, you cover with mulch between the starting plants. Most forage crops prefer slightly acidic soils, but soil that falls below the optimal range can have serious consequences including lower nutrient availability in the soil, a greater impact of toxic elements on the soil, a decrease in microbial functions such as nutrient cycling and nitrogen fixation, and overall reduction of water and nutrient uptake by plants. If you accumulate any woody residue that you don't leave on the surface, it will rot and form an excellent source of mulch for any flower or shrub you choose to plant.
All of the woody plants your forestry grinder removed left roots that will sprout with multiple stems each unless sprayed with herbicide or trimmed to stay low. But if you're going to use mulch to control weeds until you plant grass, then I doubt you'll need straw on top of that. The problem with planting lawns now is between your nutrient and water needs and the heat, many of them just won't work well, so you'll probably waste quite a bit of time and effort on seeds and caring for the new weed, with an incomplete result. Restoring and adding native species plantations and additional wooded areas surrounding these fields can also help.
The other option is to let it decompose in place for the rest of the summer, and then let it sink in the fall before planting the lawn. You can rake the mulch, plant your shrubs, and then add some mulch around the bushes, just keep it away from the trunk or stems several inches and don't stack it too thick. If you still have some weed pressure even after your control measures, keep in mind that the annuals you plant come out of the ground much faster and are therefore more competitive against weed pressure. It mostly decomposes after just 2 years and we've planted perennials and mulched that ugly area with nicer mulch now.
I would leave it until just before planting grass, and then you just need to remove what the lawnmower doesn't handle.