It’s all about interrupting chains of transmission. The more chains you interrupt, the more you limit the spread.
Manually, that means asking everyone who tests positive who they have been near within the last two weeks. That relies on memory, which is notoriously unreliable, not to mention all the people you’ve been near who you don’t know.
If you add the app, you add in the people you’ve forgotten as well as those you don’t know. Not all of them, only those who have also downloaded the app. But even so, it means more chains of transmission are interrupted, R is reduced..... and maybe more general restrictions can be lifted.
So yes, people will be going in and out of isolation, possibly for no reason. But that’s better than everyone being in lockdown all the time, isn’t it? As I understand it, if the person who triggered the alert is tested and it comes back negative, the people who have been pinged will be pinged again and told they are in the clear.
As for people not allowed to have their phones on them, employers could, you know, change their policies. It’s possible to have your phone on airplane mode but with Bluetooth on, so the exchange of ID’s can still take place, without the worker being able to call/text/check SM.
In order for contact tracing to be effective we need to be testing more people, i.e. the symptomatic people who are currently not deemed sick enough to go to hospital. So that means we have to get the number of infections down still further, before there is enough test capacity so that anyone who has symptoms can get tested.
If we get to that point, and we have sufficient contact tracing capacity (combination of manual and automated), then we have the possibility of both a functioning economy and keeping Covid cases at a low number.