I thought their main problem was with false negatives.
You are quite correct.
The false negative rate is high as they have a "sensitivity" of between around 50 and 75% depending on who does them. The sensitivity if you are ill and have high viral load is much higher (over 95%), which is what swing the decision making around them - very useful to pick up the most infectious people.
Sensitivity is the liklihood it tests positive if you do have the disease.
Specificity if the opposite: the probability it says you don't have it when you don't have the disease. This is very high with this test (99.7%) meaning a false positive rate of 0.3%.
The problem comes when you multiply it up hugely by a massive number of people doing the test, and you need to take the base rate into account (the actual prevalence of a disease on the population). When there is high prevalence (like when something like 1 in 4 people had coronavirus in London), then the chance of a random person having a positive test result would be 25%*75% = 18%
The chance of a random false positive would be 0.3%*75%=0.2%
Now, if the rate dropped hugely as we controlled the pandemic to more like 1% of people things become trickier.
The chance of a random true positive test is now not 1% but only 75%*1%=0.75%
The chance of a false positive is now 0.3%*99%=0.3%
So if the population prevalence drops to 1%, the chance of a positive lateral flow being correct is just over twice the chance it is wrong.
The further the base rate falls, the worse this balance becomes until it flips. The latest stat I can find for England is an estimated rate of 1 in 340 = 0.3%.
So
True + = 0.3%*75%= 0.2%
False += 0.3%*99.7%=0.29%
So by my shakey maths we've reached the point at which a positive lateral flow is more likely to be wrong than right. I'm tired and just finished my shift a couple of hours ago, so apols if I've misremembered and fucked all the above up