Good post. You put some thought into it, so I thought it warranted a response from someone. For the most part I agree with you, but something to consider:
"Would the organization have noticed a difference in outcomes if they just sacrificed the picks used on Biggs, Percy and Finn...?" When doing an analysis like this, I think you need to consider opportunity cost rather than a simple "we'd be in the same position". At the extreme, the opportunity cost is the best available player that was not drafted and therefore most teams fail in maximizing the value of their picks in this scenario. Without going to that extreme though, I think a reasonable case can be made that if there are a number of better players available and you select someone that is inferior (if I recall correctly there were a lot of people questioning Burke's moves), then yes, these picks could be seen as having a negative value, but that is dependent on how you define value. I think you must also consider the cost of the other asset traded to move up to make those selections (I can't recall if it was Biggs or Percy, but I recall giving up future draft picks to select him).
If we are talking absolute value, then a player likely does not have true absolute negative value (unless say he did something like Danny Heatly, in which case he cost your team a valued resource and therefore must have negative value). Note: not trying to be insincere here, that was truly tragic, but it is one example where I think we all must conclude that the draft selection had negative value to the Thrashers organization. You can look at less extreme examples like Patrick Kane, Ryan O'Reilly, Evander Kane, etc. and weigh their productivity in the NHL versus the harm they have done to their organizations, community, teammates, etc., but overall I tend to agree that from an absolute perspective it would be rare to have someone have negative absolute value.
From a relative perspective, which is what I think the other poster is really focused on, then you can easily conclude that a player has negative relative value. For instance, imagine Buffalo selected Rasmus Sandin instead of Rasmus Dahlin. On the absolute scale, both players will likely contribute and be productive in the NHL. On the relative scale though, Rasmus Sandin is negative value because he will be worse than Dahlin and therefore was a waste of a resource (draft pick) that should have been better utilized.
I would also say that despite even the best effort to be a meritocracy, there are almost always going to be failures. For whatever the reason coaches have their favourites and if playing a favourite results in overlooking a player in your organization with far more upside, then an argument could be made that you are expending resources (ice time, opportunities, etc.) on a mediocre asset at the cost of not developing the better asset. (I think this is where the other poster had some concern). This would be negative value from my perspective too and Kappy kind of fits the bill. The greatest part of Nylander not signing for so long was that it opened a spot for someone that I feared was starting to be overlooked in the organization and was possibly at risk of being relegated to a role lower than his skillset would permit.
Anyway, don't feel obligated to respond... just thought you might appreciate a slightly different perspective on things.