By far the most helpful thing for us was applying for and getting DS1 an EHCP that meets his needs, and includes therapies.
I would have liked staff to have more training on trauma. I feel we may not have reached a point where DS1's MS placement brokedown if staff had more knowledge on how to handle his MH difficulties that were rooted in trauma.
More generally, in England, LAs have a statutory duty to provide education to CSA pupils unable to attend school for medical reasons, including MH reasons. This is the guidance. If the LA refuse you can force the matter by beginning, or threatening to begin, Judicial Review proceedings.
Parents should also consider applying for an EHCNA (or statement of SEN in Wales & NI or CSP in Scotland).
Regarding your points schools must have a SENCO, SEMH needs are consider a type of SEN, so it would be reasonable to expect the SENCO to be the contact.
The school sound terrible not keeping in contact. I wonder whether part of that problem is because DD was Y11 and, disappointingly, there wasn't an incentive to keep in touch because she wouldn't be their 'problem' much longer. Sadly, they wouldn't be the first school with this view. If this was the case it is infuriating.
Education for those unable to attend school is the responsibility of the LA, not the school, so your last point would be best directed at them. The LA's offering must meet the needs of the pupil, so a different arrangement should have been offered. The provision should have been more than what you got if DD could manage more, even if it was 'just' a maths lesson each week rather than alternate weeks.