U.S. strategy in the Vietnam War has been roundly criticized for overemphasizing conventional threats (the NVA) and meeting them with conventional methods, while their strategic position in Vietnam was being undermined by unconventional means (the VC infrastructure), which required other methods to deal with (counterinsurgency, civil actions, etc). The Phoenix Program to me looks like an exception to the rule; I'd be interested in hearing how it fit together with everything from our Vietnam experts.
The elimination of the infrastructure of the insurgency is a basic counterinsurgency strategy. By decapitating the planners, financiers, commanders and everyone in between that have an impact on military and political matters of the insurgency, you throw the structure of the insurgency into chaos which you can then exploit. By capturing someone who actually wields some sort of power, you can gain valuable intelligence which in turn can help you find other insurgents. This is the basic thinking about the strategic value of something like this.
On the other hand, many critics (usually not familiar with COIN) would say that this is essentially an assassination campaign against civilians and yes, they would be partially correct. The use of the term "elimination" implies the outright killing of an individual, but we're talking about acts of removing the particular insurgent from the structure through any means possible. This doesn't always mean death, but also capture. Less than 35 % of all "eliminations" ended up with the individual in question being killed and most being captured and incarcerated by the South Vietnamese government (even though they could find themselves being released shortly depending on where due to bribes or fear for communist reprisals).
Now, that answers the question whether or not it was strategically sound. In the context of a counterinsurgency, it is a necessity. If this is the case, then how did it work in the context of the Vietnam War?
For different reasons involving incompetence, unreliable intelligence, no effective co-operation between the US and South Vietnam, the post-1969 use of the body count system as well as corruption, the program was not as successful as it could have been had it been carried out earlier and done by a government not as focused on conventional warfare as the US was in South Vietnam. Despite that, even the North Vietnamese admitted that the Phoenix program had a great impact on their influence on rural South Vietnam.
Yet it is the transgressions of the program which is worth looking at, in my opinion. Without proper co-operation and control, transgressions will happen and innocent civilians will get killed by individuals who for various reasons wants to see them dead (whether it be a grudge or the eradication of a rival). The fact that torture was used on both "guilty" and innocent civilians is also a matter worth mentioning. Although the program did have an effect on the VC, one could ask the questions on whether the end justified the means. As we all know, the United States left South Vietnam in 1973 and the VC was still up and running at that time.
In summary, the program was effective although with plenty of flaws that were inherit in the counterinsurgency carried out by the United States in South Vietnam. Without a proper COIN strategy in which Operation Phoenix could be contextualized in and work together with, it sticks out like a sore thumb and the fact that it wasn't functioning properly with other intelligence agencies in South Vietnam meant that the results were of varied success and failure.