I do not think that that method will drain and refill the torque converter.....the physics are not there for a closed container.. that is why they had to put the drain plug on it...
Since we do not know the cause for sure of your shifting problem.... to rule out the deposition of varnish from the old trans fluid is premature...
I still recommend... no matter what else you decide.... that you put Trans-X into it for a reasonable period of use... if you read the search results I posted you will see that I am not the only one with praise for Trans-X....
How would they change the filter in the trans if they don't drop the pan ? (EDIT EH beat me to this one while I was composing..LOL ).. Lots of information can be gleaned from looking at the pan inside also...
If you are inclined to do this stuff yourself I would join the others who recommend it... you will know a LOT more about your car... not only the trans... but you will see stuff when you are down under that you might need to address at some time.... and might catch other problems in their beginning rather than them stranding you out on the road...
I have never seen a "quick anything" place which was good about being able to see other things than their speciality going south...