It varies a lot with the complexity of the scenario. One tactic I use to combat this is to try and recruit as many playtesters as possible and also to playtest each scenario myself with various playtesters. The value of a playtest from anyone is important, but I try to give each scenario at least one playtest between two top tournament players, so that it can get a good, high level test. Between playing them yourself multiple times against various opponents and matching people carefully to give them at least one try, you never let one go until you are satisfied with the results of several playings.
For instance, one project I am involved in, the SCW playtests, we had roughly 8 of the scenarios definitely "put to bed" and a further 4 " very close" with another 4 still developing. Rather than let the "put to bed crowd" go, I asked some top tier players in Jersey to please give them another playing. A couple of the scenarios came back "don't change a thing" but others did require additional tweaks. Therefore, yes, you still play the crap out of these scenarios because who wants to turn out crap? Your name is one of the ones that goes on the product, so you take pride in knowing when it is fully baked.
For BFP's Sangshak, for example, the 10 scenarios I feel are currently in the bag I still farm out to playtesters, some of whom play them multiple times. Others do not. Also, Martin has been kind enough to look at including some of them in his tournament over in England this coming year, to get additional testing there.
So, to sum up my response to the original post, hell yes you still test the crap out of them. Due to our little obsession being so nuanced, even then something may slip through, but certainly anything I am currently involved in sees some playtesters willing to play a scenario multiple times and others who are not, but YOU should play your scenario multiple times before it goes out the door. You should invite players of all skill levels to play them, so that you get a feeling of where the balance is on a given scenario at different levels. While I never discourage playtesters who are new, I do understand their limitations. To expect a new player to find wrinkles in a scenario a veteran or elite player will find is simply unrealistic. Can the new player find things? Absolutely, but it is a different type of feedback. Often based on a fresh faced view of the hobby and the scenarios in general. One group helps you keep things fresh the other precise and balanced. You come to learn who you can trust to play scenarios multiple times and who you can't rely on to play them even once and adjust accordingly. Taking an attitude of "well, they will only play it once before moving on to a hotter chick" shows a lack of pride in your work. I have not encountered anyone in the last two decades who has taken that approach. I would not feel comfortable working with someone who did. People purchase modules and scenario packs. Not playtest kits. ;-)
Just one man's opinion.