If I can offer an alternative perspective I used to be a qualified lifeguard (too old and fat these days) and an a watersports instructor.
Based on what you’ve described it sounds like the swimming teacher and lifeguard did the right thing in regards to the incident and the risk of secondary drowning was low as she wasn’t at that stage yet.
Where I would be asking questions is why she was in such deep water, with the swimming teacher more than a step away (if the teacher was having to throw stuff they were too far away)
Why didn’t school call you, I wouldn’t expect them to call immediately if she didn’t need medical attention, the teachers are probably a bit shaken themselves, and had a class full of shocked / hyper / upset children to deal with as well as looking after your daughter, but once everyone was back at school safely they should have phoned.
How are they going to ensure this can’t happen again?
If a lifeguard entered the water the swimming pool will have recorded it as an incident so it would be entirely reasonable to ask to see the report and what steps they are taking to mitigate the risk, they may also have CCTV footage that you can watch, if you want to.
I personally would be asking the same questions of the swimming teacher, how was this able to happen and what can you do to stop it happening again.
As a side note - can you get your daughter back in the water at a fun session so that she can have a positive experience? Kids who have an unpleasant experience (capsizing a dinghy or falling out of a canoe) seem to do better if you can follow it up with a positive one soon after.
I wouldn’t set too much store by the children’s retelling of what happened, a couple of times people have been “trapped in canoes and nearly drowned” when the reality is that they fell straight out and popped up within a couple of seconds when people panic and get the accompanying adrenaline rush it can skew their perception, particularly of time.