@jojojo Why wouldn't test & trace exist? It is instrumental in Finland, the app has been useless, and we had one of the highest amounts of downloads.
Test and trace relies on a timeline between positive test and contacts informed.
If you go with a picture of day one - symptoms, day two - test, day three - result, day 4 - trace start phoning the contacts. Those contacts will already be in their most infectious phase and some will have symptoms themselves. Realistically, it's about which people does the person who tested positive warn themself.
In honesty that formal timeline is itself optimistic in terms of what typically happens. If you look at UK testing data, fewer people get tested at weekends, and on holidays - because most people don't react the instant they get symptoms, and presumably decide that they'll get a test if they still feel bad on Monday
If you go into the idea of asking the people who are contacts of the contacts of the person who tested positive (which you need to if infected to infectious is 4 days) then with UK case rates you'd be quarantining hundreds of thousands of people per day. It's just not going to happen here.