These tests are going to be really problematic if they have any false positive rate that is more than negligible. Who's going to carry the can if you're denied entry to the airport on suspicion of coronavirus when you're not ill with coronavirus? Will airlines rebook, or refund? What about hotels?
If the passenger is made whole (financially) then that's less problematic (not nearly zero), but if the passenger is left to carry the can then will people risk any serious cost - or just continue to give up on travel?
Heathrow says "medically grounded", which is nice - we don't need any ineffective quackery. But if you administer a test to many people, few of whom are actually positive, you will end up with more people affected by false positives than actually prevented from travelling due to a true positive.
This problem already happens in anti-terrorism, as terrorists are few and far between. If coronavirus sufferers are also rare, false positives are a serious problem.