You are not doing anything wrong. You just have a very light, well drained potting soil. The last thing you want is a heavy, compacted soil that won't grow the the healthy plats you seem have.

When transplanting it's not good to have layers (or a large clump surrounding the entire root mass) of different types of soils. Different soils/soil mixes have different properties and conduct water through them at very different rates. Sometimes this will pull needed water away from the plant and sometimes this will actually prevent excess water from draining from the root zone and cause problems. It's far better to shake most of the potting soil off and have as much consistancy whether transplanting in a bigger pot or outside into native soils.
You want your plants to grow their roots through the new soil-not be confined by them.

Also, I've been "rinsing off" roots with water when transplanting from old clay embedded soil to fresh new soil. Not only does anything like fungus or parasites get washed off, but the roots get a chance to establish right away.

Even with regular potting mixes the soil will fall off the roots at a lesser rate as roots develop. For me this is a important factor for planting acres of transplants with a water wheel transplanter where the planter has a second or two to slip the root ball brom the pack and into the planting hole. I suggest you simply give your plants a few extra weeks to form more roots. Idealy the roots will not be such a mass that they are all you see when transplanting. Avoid washing the roots as you will likely just cause them to dry out and most fungal, bacterial and viral diseases will not be washed off.

Actually, I rinse off roots or soak plants to romove the old dirt in the same situations as flysky too. As long as you don't let them sit out and dry excessively there is no harm done. Viral and bacterial disease won't be rinsed off but insects and fungus can be and it gives a chance to do a good insepction.