The signal/noise ratio will not be enough to make it helpful. Logistics on undoing a false positive test result, after notifications have been sent out, well- people are screwed because a false +ve can propagate far unless you are known hikikomori.
I agree with this, and as said earlier, had already concerns about how/why infected people would declare it through the app to begin with. We still have no idea. It sounds like something responsible to do? But there could be also myriads of reasons for not doing so.
And then, even if enough people do this, there's the false detection ratio as you said.
Another related thought - if someone is tested infected, IMO they should NOT get out. As I remember, I think in South Korea, people tested infected would be quarantined - either at home if this is possible (and a quarantine is not just confinement, you don't get out at all), or in hotel rooms if you can't stay at home. I admit I haven't really completely understood how the tracing app worked in SK, but I know they had this quarantine thing.
If we just allow infected people to move around freely - even if there is a high enough probabilty of tracing whoever gets in contact with them (obviously not 100%), this may be questionable.
Yet another point - by the time someone would be tested infected, they may already have been for a little while, without anyone being able to know.
Many dark spots in this thing IMO.
Again it seems to be inspired by some asian initiatives such as in SK, but, while I admit I don't completely know what exactly what the system in SK, I'm sure it was a lot more effective (but a lot more problematic for privacy too), and just one thing among a lot of measures. For instance, I've seen over there, they would disinfect public transports several times a day, something that's probably not possible in many other countries. As to the app itself, it did report everything to central servers so that people could actually look at maps in advance, seeing which spots could be more or less risky to go to. That's prevention, and more effective than just getting out anywhere waiting to be close enough to someone. But yeah it poses major privacy issues.
All in all, the benefit/drawback ratio of this app thing is largely questionable IMO, especially in the way we are thinking of implementing it in the West.
And deeply embedding contact tracing in OSs for future use is a privacy concern, nothing tells us it will be only temporary (it likely won't be).