My dd had similar experience when she started primary - very near to the start, her teacher told us she'd need speech therapy and advised us (no option to refer us, I had to do that myself) to get an appt straighaway.I was a bit offended, like you, as she was my PFB,but the teacher was right - because she'd largely been at home with people who knew her, we could all understand her perfectly as we were used to her (simplified and unclear) speech. But strangers who didn't know her couldn';t work out what she was saying.
She had her first appt around xmas and only needed that one - she would sit in her room saying strings of words with 'z' sounds in as she couldn't manage the letter z and two of her friends at school had names that began with it! :)
The speech therapist then signed her off.
But the school was absolutely right to suggest referral - as I was around her all the time, I would never have been aware of what she sounded like to people not used to her way of speaking.
Re the water etc - this is not a sign of a bad school, just a normal part of there being 30 kids and only a few (3?) teachers/TAs. It's hard for all kids to get the hang of rules at schoool re water, toilets etc and most kids have a few slips till they get the hang of them; this could happen at any school.
Re contacting the school with your concerns, the normal procedure at any school is to speak to the class teacher first - either wait after pick-up (after drop off she'll be busy with kids) or if she's too busy then, then make an appt for a chat with her or with the office (unless you have a direct email address to do this on). The only reason to speak to the HT is if you have already spoken to the class teacher and not got anywhere - it is extremely rude to speak to the HT ie going over the head of the class teacher without asking for their perspective on your worries first - and is unlikely to endear you to the class teacher in future.
You do seem to be letting one incidence of shouting affect your whole view of the school - shouting at reception kids is clearly not on,but that does not mean the whole school is dreadful this is a typical incident. My dd had a v shouty teacher who taught reception for a couple of years, but was complained about lots and has since left - but that does not mean the whole school was bad, just that one teacher was, and everyone in the school (incl the HT) knew about it and were doing their best to get rid of said teacher.
Finaly, just to set your mind at rest,my pfb was topp of her class academically all the way through primary and is now at grammar for secondary; she enjoyed primary lots. Having slow/unclear speech need not be a marker for anything else at all or affect their education/happiness in any way at all. Another boy who she was at primary with who needed speech therapy all through primary is no the local lothario at secondary! onto girlfriend number 6 at least after 1 year at secondary! So having unclear speech or speech therapy need not hold your dc back in any way!