There are things we know about Elizabeth Weir. She's a doctor of a social or political science (most likely a Ph.D. of International Affairs/Relations) and that her career includes the following in the Stargate universe, previous to joining the Stargate program: brokering five of the most influential treaties of recent history as a negotiator, including in the region of North Africa.
As a consequence of this, when the newly elected President Hayes was seriously considering publicizing the Stargate program he recruited Elizabeth to handle the command and running of the facility. Her first and immediate action was to shut down procedures in order to do a full examination and overhaul of the policies in place complete with the autonomy to make any changes she felt necessary.
There's a lot of reasons Hayes might have made this decision but the most obvious is, as Elizabeth pointed out herself, that a secret government program run by a civilian is far more benign in appearance than one ran by the U.S. armed forces (specifically, the Air Force branch). If the Stargate program became public, which notably of all the personnel involved Dr. Daniel Jackson was the one most firmly against that decision, then the need for a negotiator would be absolute. In the throes of a public release the need for a balanced, calm, non-intimidating person would be invaluable to convincing the U.S. public to the well intentions and non-threatening nature of the program. The fact that Elizabeth had previous to her inclusion in the program been established as anti-military action and anti-arms is the ultimate media coup.
What this says essentially is that Elizabeth wasn't chosen to lead the Stargate program, though she was given the authority to do so, but to negotiate between the program itself and the views therein and the public (first U.S., then by necessity international). Her leadership skills weren't in consideration as a primary factor here because once the Stargate became public running the program would be child's play in comparison to convincing the public not to riot over something like this. We've tried to impeach presidents over things like this before.
Of course, with the discovery of the Antarctica outpost and the destruction of Anubis making the Stargate program public became a secondary concern to finding Atlantis and possibly the Ancients themselves. Therefore the SGC was reinstated under non-civilian control, eventually going to one General Jack O'Neill, and Elizabeth was moved to the Antarctica outpost, where we find her in Rising, and given the task of assembling a possible team for an expedition if Atlantis's location was discovered.
But why was she chosen for the expedition lead if she was not a leader, as I posit? Simply for the same reasons she was chosen as leader of the SGC towards of the end of S7.
We've known for years that the civilian funding for the Stargate program is paramount to their continued function, and that the international aspect of that funding is a major political concern as it has been for the nearly half of the program's running time. The Atlantis expedition is a result of the political quarrel in large part, and includes members representing the interests of dozens of countries, all of which are informed of the Stargate's existence and all of who have been jostling for position within the program. A struggle that was far more complicated when the primary leader of all Stargate action (whether at the SGC itself or in Area 51) was military.
By making the Atlantis expedition international and civilian the U.S. gov't is able to appease these countries and agendas with very little initial cost to themselves, as the funding would also come in part from those countries and the Atlantis expedition may very well be a failed experiment (as they had no certain idea of what they would find, if anything, on the other side). In that sense it's a great political maneuver and allows the SGC on Earth to remain in military control with far less contention than would otherwise be evidenced.
Additionally, Elizabeth's skills as a negotiator would be valuable here as well because her primary role as leader of the expedition is to make certain that things move along smoothly. And therein lies the significance of the fact she is a negotiator and not a diplomat.
A diplomat is someone who is skilled in conducting negotiations between representatives of groups or nations, someone who is *accredited* to represent a particular group or nation. A diplomat may privately want to promote peace or not, or another personal agenda or not; but their job is and always has been to promote the agenda of the particular group they represent. In an informal or social sense, yes, a diplomat is someone who uses tact and non-confrontational approaches in order to gain an advantage. But a diplomatic person is not the same as a diplomat.
Therefore, Elizabeth is a diplomat of Atlantis (but not on Atlantis), as a colony, and a representative of their agendas in the Pegasus galaxy. Or, alternatively, she assigns John the role of diplomat, though with varying success. Teyla could also be considered a diplomat of the Athosian people, as that is her primary role in regards to the Athosians. She represents their trade agenda, and as is fairly typical of tribal systems -- such as the Pegasus galaxy presents in macrocosm -- the leader is also the primary diplomat.
What Elizabeth is, of course, is a negotiator. A negotiator is much more akin to a translator than a politician. As an unbiased, third party capable of communicating the desires and needs of one party to another in order to facilitate agreement or, in many cases, convey a message because the two parties who cannot or will not meet in person, a negotiator's role is not to make judgements or propose solutions.
What she did with the Genii in regards to the nuclear devices in 'The Siege' is negotiation, communicating a proposition that one side is suggesting to the other side. In this case, the side conveying the message was the one she happened to be a member of but then again they didn't have an unbiased third party available. And the fact she was held at gunpoint and treated much like a prisoner isn't atypical of heated negotiation situations, especially in areas we know that she worked on Earth (North Africa). It's extreme, but negotiation is a horrendously dangerous career in many aspects when one an active, field negotiator.
Another function she serves as a negotiator is on Atlantis. Her job, in large part, is to negotiate between the needs of the military aspect of the expedition and the needs of the scientific aspect, to establish and maintain communication between the two mentalities of her expedition, though she probably has help of people such as John Sheppard who can equally claim foot in either side (and any of the military computer techs probably fit right in with the civilian ones).
Her job is also to facilitate communication between the different countries, cultures, and worldviews possessed by the members of the expedition, which are almost certainly as widely varied as the number of countries (and regions within countries) they hail from. Atlantis doesn't need a leader as much as they need a referee, something that she makes reference to more than once herself.
So, regardless of Elizabeth's personal views, which are varied themselves and highly situationally dependant, that's neither her function on Atlantis nor what she mostly does, or chooses to do.
This is something reflected clearly in her treatment of John Sheppard and/or Rodney McKay and/or another member of the expedition (such as Kavanaugh or even Teyla), as she's allows them very often to do their own thing unless it becomes an endangerment to the whole or utterly impractical. And in each time she has made an order which was ignored, vaguely interpreted, or disobeyed, she has taken the person responsible to the carpet in the way most suited to their particular personality.
Her role on Atlantis isn't a commander or even a leader, it's a person who can keep these people from killing each other, themselves, or innocent persons. That's all, and she mostly does a phenomenal job of it.
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All comments, thoughts, and questions are welcomed, except flames (i.e. attacks on my morality, intelligence, or political affiliation. Let's keep this un-sandbox wars, 'kay?)