He had a cough. Doesn't matter if it's due to hayfever; the school, the GP and Mrs Evans down the street can't know that without the negative PCR to confirm it.
There may be twenty two other children throughout the school who have also been kept at home due to being 'unwell', three in the same class as him who are already awaiting results or had to be sent home with raised temperatures or they know of an outbreak in the nearest other school where older siblings of children in your school attend.
Right now, if a bubble bursts, it screws up the last two weeks of term for kids and staff alike. Which will be shit for them. But if somebody's hayfever turns out to be Covid in another week when they've been in class as normal, that could mean a mass outbreak, people's holidays screwed up, losing summer childcare money, greater financial pressures just when summer holiday poverty is a thing and cause another spike in risks to vulnerable kids, staff and the wider community.
LFTs are utterly crap. Everybody who has had a positive in the last six months that I know (including somebody who ended up in hospital just a few hours after doing a negative LFT) has only tested positive on PCR. Having even 'maybe it is maybe it isn't' symptoms is a good enough reason to have to insist upon a PCR.
The result will come in soon. Nothing anybody else can say will change the school's minds because they're following the correct procedure.
It'll most probably be negative - but nobody knows that until the negative is in.