what normally happens is the git programming the hospital PABX has the option of 'present a single number, present actual callers DDI, or present nothing (default) '
I actually had think about that, but being a commercial company we opted for 'give our main number'
But lots of people never even get asked the question.
And as long as it is not an 'issue' on some 'manaqement' radar, it doesn't get (trivially) fixed.
Same thing happen when email starts getting rejected for 'invalid sender' reasons. people fix it, or bugger off. . Of course the NHS can do what it likes with your money.