The cheering squad thing doesn't add anything to the question, does it? Cheering or sitting down has the same lack of impact, correct?
Interestingly (or not), from your initial description I ASSUMED you'd pause and re-acquire opponents for everyone between each contest, so it would be fully single-elimination. Your variant where everyone matches and plays at their own pace seems insane - it's obviously broken for exactly the reason you state: the winning strategy is to pick your opponent as slowly as possible, preferably when they're the only other competitor remaining.
Why would anyone play in such a tourney? I guess it becomes a slow bicycle race if you add that you'll be eliminated if you're noticed to be slow-playing. But then it's a weird game where part of the rules is that not everyone can be aware of the rules.
The cheering squad doesn't affect the rules, but has been part of this every time I've been in one of these team building exercises because of, well, team building.
Ah, I'd never seen this at a conference, and I _have_ played in real-money rock-paper-scissors tournaments (which were strict single-elimination, with random assignment), so I figured it was some proposed mechanism for actually deciding something.
As a team-building excercise, there's an additional complication in that the definition of "win" is debatable. If there's no prize and no consequences of early or late elimination, I suspect my utility would be improved by simply not trying to optimize anything, or perhaps by just using https://tvtropes.org/pmwiki/pmwiki.php/Main/PoorPredictableRock and being "out" as soon as people realized I was serious.
Let's say you're at a conference and the leader gets everyone to play group Rock, Paper, Scissors as an icebreaker:
How do you maximize your chances of winning?
This is structured as a single-elimination tournament, except there's nothing to ensure that it's a proper binary tree. Because different games will take different amounts of time, and you're about 50-50 on winning each match, you want to play as few matches as possible while your opponents play as many as possible and eliminate each other. It's like a Slow Bicycle Race, where everyone has to keep making forward progress, but only just barely.
(You can also try to win by predicting your opponent's choices, but that's very hard. Unless your opponent is a small child in which case an "always choose paper" strategy can work well.)
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