Answer: In my experience all three can do equally well. Modern nursery practices have made it possible to produce good quality plants all three ways.
In my opinion, to a great extent it depends on the care the plant receives in the growing and shipping and retail handling processes that determines its condition when it reaches the final planting site. If there is a lapse in care that stresses the plant, then it will perform less well after planting.
Bare root plants are limited in that they should be planted in earliest spring while still dormant, the others can be planted any time the soil can be worked -- meaning not frozen and not too wet.
With container and B&B plants, I like to plant in the fall when roots can become established during a cool and usually normally wet season and still have more time with gentle weather in the spring before the heat stress of summer hits. However if that means the plant has been sitting around in a retail environment becoming shopworn and suffering heat and drought stress all summer then it may not be a good thing. The wholesale grower will have the equipment and expertise to keep the container plants healthy all summer, and would leave the B&B plants in the ground until close to shipping time.
Also, planting technique can have an effect. For example, bare root plants should have their roots soaked in water before planting. B&B plants nowadays tend to have synthetic wrappings, all of which need to be removed to allow free rooting. They also are sometimes dug in a manner that leaves them too deep in the rootball so that needs to be remedied at planting. And container plants need to have their roots spread out or sometimes even cut if they have begun to grow in circles inside the pot.
I think establishment period depends on the condition and planting as above, plus the aftercare such as careful watering. The site itself will affect the length of time as well, not just the soil conditions but also microclimate and wind for example -- not to mention the weather the year they are set out. If planted in a drought year, even with watering, the plant may be slow.
Also it seems to me that the size of the plant has some bearing on length of time it takes for it to become well rooted and kick into growth where a younger plant usually seems to be less stressed by the move. The rule of thumb with trees is it takes a year per inch of trunk caliper. Often enough you will see a whip planted the same year as a more expensive larger size catch up within a few years.
There will of course be some differences among specific plants, but this is generally how I see it. I hope this helps answer your question.
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