Not true on two counts:
1) Championship Play is defined as 40 minutes at the top of the first column on p.5 of the Tournament Rules, although granted the text at the top of column two has not been updated to match and still says 35. Since the 40 minutes is in blue it represents a deliberate amendment from the previous version, so I think we can safely assume that 35 minutes is an oversight that FFG missed in editing.
2) You don't have to win in the time limit, you just have to be ahead after time is called and both players have finished a turn.
Irrespective of those facts, having a choice is important in swiss. For example, if I only need a single win to make the cut and I have a favoured match-up I'm more likely to select it first to give me the luxury of time to leverage my advantage. If I'm paired against a random player who I'm not familiar with then I am way more likely to run second (especially if they're repping one or more "slow" IDs) - the reason for this is that I feel I can adjust the pace of the game a lot more as the Runner, so if I'm playing someone who I don't necessarily trust to play quickly and confidently (i.e. I suspect we might go close to time) I'd prefer to be in a position where I can have more of a say in moving the game along faster when it matters. I think this makes more sense than rushing unnecessarily in the first game and risking an error in order to buy time for a slower second game. The state of the meta makes a difference as well; for instance if the Corp was heavily favoured in general then I would want to play that first to register a safe 2 points before punting my Runner as the underdog. Lots of factors go into the decision. Sometimes it can be something as simple as the fact that I just played Runner, so my head is in the Runner groove and my deck is already out of the box / shuffled. If the rules say I have a choice then I'm going to exercise that right.
At the end of the day I don't much care what the rule is, I just care that it is consistently enforced. The original poster wrote in the report that he made a decision to choose a side in the second round of double-elimination in order to maximise his chances of getting his favoured side at the business end of the tournament. Absolutely smart play on his behalf, but ultimately not a decision that he should have been allowed to make because the selection should have been random. He potentially gained a material advantage over other people in the field who determined sides randomly and ended up getting their least-preferred side later on.