I wouldn't say it's any lack of competence on the part of the staff - because it's very likely that they have been on the receiving end of formal complaints for not sending home children who were distressed, anxious, not quite right in themselves or blatantly just wanted to go home and knew that all they had to do was say they had a headache/tummy ache/felt sick/whatever and Mum would come and get them.
It's very frustrating when you're sure that there's nothing physically wrong with them and they should be encouraged back to lessons, but are equally sure that if you don't send them home, you'll be subjected to a rage with the power of a thousand burning suns for not 'caring' or believing their child who would never, ever lie. And some children, NT or not is irrelevant, can be very adept at the Dying Swan act so that they really do appear to be genuinely unwell until they get what they want Mum arrives, at which point they stage a recovery worthy of Lazarus himself.
Actually knowing they have your support to keep the DC in school is vital - which is better done through her teacher, as whoever is responsible for making the calls often doesn't have the authority to override a well meaning class TA or teacher who has already told the child they're going to be sent home - worse when lessons are taught by different staff, as somebody sent back once would then just repeat it all to the next teacher they saw after they'd eaten lunch/played/done the lesson they enjoyed, but also done with the same teacher by escalating; like the fifteen curtain calls when a child doesn't want to go to bed, wanting a drink, a wee, a biscuit, they're hungry, they're cold, the teddy is in the wrong place, another wee, something really important happened at school, what would win in a fight between Godzilla and Lightning McQueen, there can be headache, tummy ache, toe hurts, leg hurts, I bumped my head, fake sick noises in the toilet, limping on the opposite leg to the one they limped with five minutes ago..oh, cough, cough.....
In short, if everyone knows the procedure is to encourage them back into lesson unless something is hanging off, broken, bleeding profusely or they've puked all over the carpet and Mum agrees with this approach, it's a lot easier to follow it than trying to second guess whether Mum will go mad about a) not sending them home immediately, b) not calling the instant there's a weak little voice saying my tummy hurts, c) suggesting 'let's see how you get on for a bit' and seeing they're fine once they've gone back to class or whether she'll be angry at you for calling and disturbing her in the first place - or if she'll support you and if it turns out they are genuinely ill, won't have at go at you for misjudging it (usually that's when they go on to puke spectacularly in the doorway of the class about thirty seconds after being encouraged to go back).
[Yes, before I get pulled up on this, fathers can be involved too, as can grandparents, but 97 times out of a hundred, it's Mums because Dads won't pick up the phone or are too busy and important and grandmothers practically always want them to go home because they get some extra time with the children that way]