My personal take on how it works and why they used the phrase "based on teamwork rules", which I will fully admit could be wrong, but here's what I think the RAI is:
Teamwork and SUT work in a nearly identical way mechanically speaking. Lets use an example of a 4 man team:
1. Team agrees that person A is designated as the leader, and persons B, C, and D are not the leader. Under "teamwork" rules (SR5 p.49) these non-leader people are called "Assistants", whereas they are simply "team members" under "SUT" rules (RnG p.98).
2. Each assistant/team member rolls their skill roll and counts up successes as normal. This is exactly the same for teamwork and SUT.
3. The designated leader the rolls his skill per normal but gets a number of bonus dice equal to the total number of hits made by all his assistants and gets a limit bonus equal to the total number of assistants who rolled at least one success. This is exactly the same for teamwork and SUT.
I imagine the reason why they used the phrase "based on teamwork rules" is because this is basically exactly the same as teamwork rules. They simply go on to reiterate and explain them in slightly better detail. I feel the intention here is to view SUT tests as nothing more than a codified example of a specific type of teamwork test with specialized outcomes/results (the maneuvers).
This interpretation makes the rules consistent, simple, and unambiguous. It does, however, also mean that the sidebar on p99 of Run & Gun should be revised by making swapping the positions of steps 2 and 3. I'm sure there are other interpretations possible as well, but this seems to me to be the easiest and most straightforward interpretation.
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As for the 'initiative issue': An SUT maneuver only lasts for 1 combat round. If you want the bonus next round, you have to re-roll a new SUT test. That part seems pretty clear and simple.
It doesn't state exactly how this works, but based on the fact that each person has to make a roll, and therefore has to make it on their own initiative pass, I would assume that the leader needs to make his roll 'last'. This doesn't mean that anyone must delay their actions, it simply means that they won't get the bonus for any actions they take until the leader makes his roll. I've been in tactical situations and coordination doesn't always happen simultaneously, so this 'staggered' actions thing totally makes sense to me from a realism standpoint.
Anyway, each person makes their own SUT roll as their free action on their initiative pass, and then, after everyone involved is "ready" (has made their SUT roll on their respective initiative passes) the leader says "now" (makes his roll - starting the maneuver).
The GM should make a note of the initiative score/number that the leader's "maneuver start" roll occurred on because the bonuses from the maneuver will be in effect for the rest of the current combat round (all remaining passes/actions) and the beginning of the next combat round (kind of like how grenades go off on the next combat turn on same initiative score they were thrown on).
Note that since a new SUT roll has to be made each round no matter what, if a team were savvy enough they could "sustain" a maneuver. This is nothing more than simply making a new SUT maneuver roll for the same maneuver before the old one wears off.
After a team makes the initial SUT maneuver roll, one of of 2 things happen the following combat round:
1. If the team doesn't want to "sustain" the maneuver, they just don't make their SUT rolls and play continues on as normal. They still receive the maneuver bonus on all actions they take up until the initiative score the maneuver was started on (last round) comes to pass (kind of like a grenade going off), at which point the maneuver bonus has run it's course and goes away. (EX: a maneuver is made on 13 of this round, everyone goes and does things on their passes with the bonus in effect, and eventually a new round is started. Everyone that has an action before 13 still gets the bonus from the maneuver from the previous round, but anyone going on or after 13 does not get the bonus since it wasn't "sustained".)
2. Assuming the team wants to "sustain" the maneuver, everyone on the team makes another SUT roll on their own initiative passes in the new round, the leader can then make his roll when his turn comes up. Note that people don't have to make their SUT roll/action on their 1st available pass since the maneuver may still be in effect from last round. However, everyone should try to make their rolls as soon as possible so as to allow the leader to make his new "sustaining" roll before the old maneuver "wears off" to prevent any "gaps" in the maneuver. If, for whatever reason, the bonus from last round wears off before the team leader can make his "sustaining" roll, then any/all actions taken during this gap don't get the maneuver bonus. (EX: if the maneuver was started at 12 last round, but the leader doesn't get to make his new SUT roll until 7 this round, then anyone going on 12, 11, 10, 9, 8, or 7 this round doesn't get the bonus.)
So, by consistently making SUT rolls each round, the maneuver can be "sustained" indefinitely so long as people keep making their rolls and the leader gets enough hits to keep it going and he gets to make his SUT roll each round before the old one wears off. Occasionally there will be gaps in the maneuver - just like in real life - nothing is perfect.
Again, all this makes perfect sense to me and makes the SUT rules very consistent with other mechanics that are similar (namely teamwork and grenades), but in the end this is just my interpretation and I could be wrong.