I think lack of reporting is probably a good number of the missing tests.
For me, it's a case of waking the children up earlier twice a week. Ds normally gets up and does it, dd gets up last minute anyway. Normally ds does it in time to report both to school and government before we leave, but not always.
With dd we're normally popping it in the car so we can check before they go in. I've had a number of times where I've thrown it away from the car without thinking, or just forgotten to register it.
I've got 2 older teens who do the test with no fuss and get it done, but still there's days where one or more of the registers (either for school or gov) get forgotten. I don't have data on my phone, so I can't register it in the car as she goes in. I suspect if you have reluctant or younger secondary school age who needs help, then doing the test is far more important than registering it.
Also (purely anecdotal) the number of tests tends to increase when there are cases around. I know our secondary was dropping percentage of tests over last half term-they hadn't had a positive since December. They've had 2 positives this week, and I believe the Thursday test was well up.
The opposite happened in another local school. They had some positives back in March and were getting a really good response for doing the tests. Then it dropped again as they didn't get more positives. They then got another positive at the start of May, again went up for a couple of weeks, but now has really dropped again.
This could mean that the positivity rate is higher than it should be.
I don't see why the gov.uk reporting system couldn't have been the only one. They ask what school, so it would be fairly simple to have a system where it collates results and emails them to the school each week. I presume they contact them immediately if there's a positive anyway.
Monday: 344 tests, 0 positive, 340 negative 4 unclear
Thursday: 562 tests, 1 positive, 559 negative, 2 unclear
Is what they need, plus details of the positive tests.
Our school form is only for filling out if negative-you contact the school by phone for positive, and redo for unclear.