So there are two types of tests PCR and antibody.
PCR tests look for the nucleic acid of the virus (the genetic code) and will indicate you have the virus right at that point in time. PCR tests are usually accurate but require a certain level of technology to perform. Hence for everyone to have a machine in their house would be very expensive... And, the test would be simple enough for people to perform, remembering the average reading age of an adult in the UK is 11, and most people can't wear a mask properly!
Antibody tests can be cheap - indeed the "pregnancy" style kit is cheap to manufacture lateral flow device.
Antibody tests can tell if you have got the virus but also if you have had the virus. You typically have one type of antibody in the first stage of the infection and then another antibody as the virus takes hold.. I won't bore you with the science.
What it does mean is that you could test positive for an antibody but no longer infectious.
The challenge with antibody tests is the level of sensitivity required. When people catch the virus they have varying levels of immune response, leading to different levels of antibodies, and in some instances people can can the virus whilst not expressing high levels of antibodies, or antibody expression may only ramp up over time..
So there isn't a magic bullet at the moment. I fear that the comments made by Johnson were to hide the mess that is currently happening with track and trace, and the lack of tests at the moment.
When Johnson was talking about tests he seemed to skip between PCR tests and antibody tests messing up the two types of technology.
To be honest his smoke and mirror approach works as we are talking about this and not the millions paid to Dido Harding (married to a conservative MP) or the millions wasted on PPE that didn't work (and again channelled through an advisor of Liz Truss.