I am always curious to hear other people's ways of doing things.
I would agree with Skopje that timing is really important with watering after you really manhandle the roots.
With respect to soil replacement, I am kind of lucky in that I am able to choose the soil from day 1... so there's no need to get rid of somebody else's soil most of the time. It's only after 10 years or so that I start seriously considering a total soil replacement for the mix that I've gotten used to. Mostly out of laziness, but the cocofiber is quite durable.
Also, as a matter of proportion and scale, it's worthwhile thinking about how much of the soil in the new pot after you've moved the plant is going to be new soil. Whether you're using a similar sized pot or going up a size. When I leave the root ball undisturbed as I usually do when I'm moving my seedlings along, I move the whole package to a new pot that is usually about twice the volume of the original. Even though it's only going 1 size up in diameter (4-5-6-8-10 inch), the amount of new soil roughly equals the volume of old soil. It may be counterintuitive, but volumetrically those pots are about 2 fold apart.
If you want to bare-root a plant in no time flat, you can put it under running water, and you'll end up clean. Maybe it's not a good idea for some sensitive plants, but for agaves and fat-rooted plants like that, it works. You just have to be extra super careful to let the whole show dry out completely before you even consider taking the next step.