This kind of centralised approach could revolutionise annual vaccinations for flu. Not sure how the rest of the world handle flu, but in the U.K. elderly people are vaccinated for various strains of flu every year - all based on prediction. Having accurate data on the specific strains and where they are spreading could, again, be a game-changer.
It is not just the elderly who are vaccinated every year, but also those with various health conditions, so it is quite a lot of people of all ages. But I do not see how this would help with the flu.
The only way public health authorities would know which strains are most active are when people get so ill they need to be hospitalized and tested. That testing would reveal which strains present the most threat, and which areas are most at risk. The only value the app could have is in telling those who have been near an infected person to get tested and or vaccinated, and maybe prioritizing areas with the highest prevalence.
But that system of tracking and testing would be more expensive and complex than simply just vaccinating everyone at risk, something that can be done during any routine G.P. visit or by any local pharmacist. And vaccinating people unnecessarily is safer than deciding not to vaccinate people based on their area or contacts.
Although you could tailor the vaccine given based on the reported strain, you could do that now. Either way you are waiting for the first affected people of the winter who need testing. But as a reactive measure it leaves people vulnerable until you have a large enough sample to say which vaccines to give. The app could not speed that process up. The mass vaccination programme being proactive ensures those most at risk have protection immediately.
It is W.H.O. which predicts which strains need to be protected against ahead of each season, and part of this is so that manufacturers can produce vaccines in large enough volumes. So this is another reason why you need to be proactive. If you waited until you knew which strains were most active, you introduce a further delay before being able to protect people as you wait for manufacturers to start production and can reach sufficient quantities.
The point of all of these apps is to avoid lockdowns. They do not protect people individually from the virus but seek to reduce the number of people spreading it. When someone who has the virus is tested as being infected it can warn everyone they have been in contact with to either isolate or be tested themselves, in accordance with local policies. The alternative being the current blanket measures to, as much as possible, prevent any sort of interaction.
Whilst this could be done with the flu, government would not consider the lower mortality risk as worth such a system, especially whilst mass vaccinations though imperfect (for the reasons you state about having to predict strains) have been reasonably effective. More importantly, the public are less likely to support the tracking measures for something that those without health conditions do not perceive as a threat. Especially when for the purpose of an app to provide any actual benefit would be to require people to isolate. Which they generally already do anyway when they have the flu.