I think people not working in schools don't realise how employment processes are quite different.
There is none of this secretly applying for a job, telling no-one and taking a day of holiday for the interview, and only telling the employer after a job offer has been gained, when references are then finally taken up. And none of this giving a months notice and then being gone.
In teaching the labour market is very inflexible. People have to give a term or half a term of notice. So for a Sept start, someone would have to resign by May Half Term. Quite often, people get jobs 6 months ahead of starting. There is a tight recruitment window. With the May deadline, in the couple of weeks before then, most schools can be recruiting. They need to interview and offer on the same day and require an acceptance that day. If they don't work to this timescale and have already taken references, they find that their shortlist of 5 only has 2 turn up for interview as the other 3 got jobs the - days previously, or if they don't require acceptance on the day, the candidate might get another offer the next day.
Therefore, references are sought in advance. Schools know who is looking for a job on their staff and it is expected that immediately you are offered a job, you tell your original Head. They will likely advertise or even interview immediately because the timescales are tight. If a schools finds itself with several staff getting Sept jobsright at the May deadline, they might find they have a very limited pool to choose from....no teachers in existing jons available and able to give notice and just those without jobs or newly qualified....might not be a great pool.
Schools also have very little slack in their budgets. When someone is planning to leave who has some experience, there iften isn't spare cash to ask them to stay or ask them to take on more responsibility. And there is always the chance they can be replaced by a cheaper, younger teacher which can be very welcome for a cash strapped school, however much they like the exiting teacher.
Unless you do something really unusual that's hard to replicate, classroom teachers come and go to be honest. Most schools are glad to be able to hire a younger, cheaper model.