How do we know that people who are genuinely ill will be believed? And what good will it to be believed afterwards when you have just been sick in front of all your classmates, maybe your bullies? You're never going to live it down.
The problem is that this HT seems to think of all pupils as adversaries or potential adversaries and is incautious enough to tell them so.
Chances are, every class contains a handful of pupils who are suffering far more from the badly behaved pupils than any teacher could. And a handful (or more) who are actually hard-working and well behaved but that never gets noticed. Now HT has just told them all he will not believe any of them and his default position will be to assume any pupil is lying. So he's lost their loyalty.
New HT in ds's school did precisely the same: announced a tough regime on arrival and assumed everybody was an adversary to be thought. Didn't take long to send the school into Special Measures, all the best teachers leaving and even the best behaved pupils developing a meh attitude.
Too much stick, not enough carrot. It's that politician thing of "If I sound good, then I must be good".