You do have to give them a P45 and they can work out your pay from your tax code.

You don't have to give a P45. It might mean that your first salary is on the wrong tax band, but that sorts itself out fairly quickly. See https://www.gov.uk/guidance/starter-checklist-for-paye

They also don't know if your tax band is due to salary, loans, deduction, pensions, second incomes etc.

HMRC seem to be able to provide a new employee's correct tax code almost instantly these days (part of the Making Tax Digital programme, maybe?).

I changed job last year and wasn't asked for a P45 by my new employer. Despite starting near month end, my first month's payslip already had the right tax code.

The days of being stuck paying emergency tax for months finally seem to be at an end!