You introduced me to Fate on this very site more than a year ago, and you mentioned this exact scenario to me a few times. This has resonated with me (as an avid Stargate - and now Fate - Fan) and I have since tried this out on my own. My findings below.
We decided that each character on the team should be treated as a full character, with aspects, stunts, skills, and stress. We also determined that the team itself would be an extra with aspects, stunts, and skills (but no stress). We decided that the teams skills would be used when dealing with the team as a whole, and that character skills should be used when specific characters are being challenged. The pool of Fate points is shared by all characters and the team.
To balance things out in our game (and make it easier to maintain), I decided that 3 aspects and 3 stunts per character was a bit too much. So instead I figured that a high concept, a trouble, an aspect, and a stunt per character, plus an additional stunt or aspect was appropriate. Therefore a character would have either 2 stunts and an aspect, or 2 aspects and a stunt - As well as their high concept and trouble aspects. Then, I gave each character only 15 points of skills, as 20 would feel like too much, anyway - when working as a team, they would use the teams skills instead of their own, so their skills could be more focused. I also decided that a pyramid skill system was not good for our purposes, and instead used the column skill system.
For the team I decided that unlike a traditional extra in Fate that is comprised of points from your own character, it would have its own points at the start, and as the game progressed it would balance out with milestones. In the end I decided on a high concept, a trouble aspect, 2 regular aspects, 2 stunts, and 15 skills, (so almost like all the leftovers from the other characters - if they were full characters).
I decided that the refresh should be per-character, and the pool of refresh combined. This way, on significant milestones, the characters would each progress in their own refresh, and if desired, could continue pooling for extra fate points, or take stunts, or aspects for themselves or their team.
In this case, the team took on most of the combat skills, while the characters took on more specialized skills.
And that is how we chose to interpret the Fate fractal, and how we designed our Stargate team. Some examples of play below.
Being attacked by a Jaffa patrol while dialing the Stargate:
The team decides to make a skill challenge to attempt to hold the Jaffa back while one of their members dials the Stargate. The skills chosen were shoot (to maintain distance and hold them back), athletics (to keep behind cover and avoid being shot) and of course a craft (to activate the Stargate). In this case, the team is acting together on the first two rolls, so the player rolls the teams skill in shoot and athletics, but for the craft roll, it makes sense that only the character actively using the DHD would roll and use their craft skill.
Whatever the outcomes, the player could use his Fate points to invoke aspects from any of the characters or the team as appropriate, giving the story its flair.
If they had failed this skill challenge while under fire, a conflict would have ensued...
Ambushed by Jaffa
In this conflict, the characters do not act as a team, instead relying on their own characters stats. This is why in most cases they would like to avoid conflict, because as a team the characters can use their teams skills instead of their own (and in our game, the team is better in combat than the individuals). When taking stress or consequences, they are applied directly to the character receiving them - and not the team, as said earlier, the team doesn't have stress or consequences. However - the teams aspects and stunts are always available to the characters. In my case however - a handy stunt I called "Teamwork" allowed the player to spend a fate point to use his teams skills for all his characters instead of their own (when higher) for a single round of a conflict.
The system is of course, not perfect, inter-team interaction is not very engaging, as it's hard in any system to interact with yourself in any meaningful way. So those situations were mostly resolved by simply giving their conclusions and the highlights of how the conversation must have went for the characters, and how they feel about it when it's over.
What I really enjoyed with the system is how natural it is to switch between characters, at any given moment, you will want to "activate" the character most relevant to the situation, when applicable, and so the story really feels like there are a lot of characters in it. This leads nicely into something that I only started noticing after a few games - and that is some distinct personalities on the characters, related to not only the initial story we gave them, but also to what situations they were excelling at versus others - one character getting jealous that he never gets to do X because someone else has a slightly higher X means that in the story you want to start using the character with a lower X so he doesn't feel bad. And then the characters start driving the story on their own.
The stunts were really fun to design for the team, giving an interesting twist on the normal Fate mechanics. Such as a stunt that requires each character to make a separate skill roll in order to accomplish an outlandish maneuver.
Many parts of the game made sense to do as montages, which was fun because it gives each character a brief spotlight to shine in. Instead of only using the single character who is best. Like when examining ancient ruins, it would make sense that the character with the highest lore would examine them. But meanwhile what is everyone else doing? So this gave opportunity to always see whats going on with the characters, and play around with them.
No boring bits. As above, when one character is doing some boring examination work - other characters can be exploring. As opposed to when playing with multiple players - in this case the same player is doing both, so feels no down time, but also doesn't feel like his "boring examination thingy" isn't useless, its just happening in the background.