Part of the problem is that the mainstream media - including the BBC - seem to have mislead people about how well contact tracing worked in places like South Korea. For example, they ran an article a few months back about how it was so effective that South Korea could reopen their pubs and nightclubs. In reality, they'd tried doing that multiple times and every time, cases had started going up enough that they had to close them again - but you wouldn't know that from the article. Nor would you have known that the same thing happened that time; the BBC kept on completely ignoring the rise in cases there even as they reported less important stories from South Korea.
Covid still isn't in control in South Korea. (Singapore maybe, but that relied on their society being structured in a way that wouldn't be politically acceptable elsewhere. If you take a look at the infection rate in their worker dorms, and realise those people would be living amongst the rest of the community in equally crowded housing and shopping at the same stores in Western countries - just like they were in Singapore until they were forced into dorms so actual citizens didn't have to look at them - and that those infections weren't even counted as "community cases" there...)
It's more in control in South Korea than it is in Europe or the US. The daily infections (per capita) are lower than the US or UK, even though they have not deployed nearly as much vaccine. Things may end sooner in the US, but probably not better overall (South Korea is likely to speed up vaccinations once more vaccines are globally available).
I'm also under the impression that their interventions have been less drastic than the US (but I haven't been following closely).
Are we really stuck thinking that a thing has to be a complete solution to be worth doing? It seems clear enough to me that identifying clusters and helping infected people isolate both have a big positive impact (both in disease control and in keeping things more open). Apps can be a piece of that puzzle.
China also shows that these apps (which combine track and trace, with status) are very useful to re-open the economy after lockdowns in a way that helps contain further outbreaks.
Of course, saying "it seems to work in China" is the surest way to having a course of action shot down as unacceptable whatever its actual merits...
It does seem to work here in Hong Kong. Two weeks ago someone who went to my gym tested positive, I was instantly notified through the app (as did 500 other members). Within two hours I got tested, and received my negative results the next day.
Anytime someone tests positive, the government notifies everyone who was in the same building in the past two weeks, and testing is mandatory.