I wouldn't, if only because if they're really dry they'll resist rotting, pack down and form a slick layer that might be difficult for the roots, water, and oxygen to penetrate. Better to moisten them, mix them with some greens, and put them back in the compost pile for a bit.

They're "done", IMO, when you can't tell what they used to be.

Better yet, use them as mulch around the plant after you do the backfilling.

It used to be commonly recommended to add a lot of OM when planting a tree so the tree would have good soil to get started, but that approach can backfire for a couple of reasons, so it isn't often recommended anymore. One problem that can occur is the "bathtub effect" where the soil around the tree acts like a bathtub compared with the denser soil around it, so it gets too much water. Another drawback is that the difference between the "new" soil and the existing soil can create a barrier of sorts and the tree's roots can't get past the barrier.

What kind of tree?

If it's a fruit tree then keep the soil on the lean side. This will encourage fruit set instead of tree growth.

If it's an ornamental tree then by all means mix leaf mold with soil and compost to encourage growth.

Instead of staking the tree, try an old Norwegian trick. Put heavy rocks on top of the root ball to prevent it from moving. This will allow the fine roots that go from the root ball into the new soil from breaking as the stem moves back and forth in the wind. This allows the stem to grow strong with wind movement. The rocks also work as mulch. In Norway, rocks are plentiful and cheap.