Earlier quoted context omitted.
As an Australian who has privacy concerns their government's COVIDSafe app (see https://github.com/vteague/contactTracing ), and hence not installing it, I'm really thankful that Apple and Google are pushing this model of contact tracing. We still don't know if digital contact tracing is effective in practice, but it's still important to try, but we can do this in a way that avoids giving governments with worrying au…
Having Google/Apple develop a tracking technology is the same as the US government having it. If you don't believe, read again what Snowden revealed several years ago.
This is why you want an API that never uses any data you don't want anyone else to have. That's what this API is.
The "trust" here isn't about whether they'll keep your data safe from third parties including state level actors. Your "trust" only needs to be that the API does what it says on the tin.
Which leads to this conclusion: either you A) trust this API to be what it says: something that doesn't ever deal with any sensitive data. Basically an exchange of random numbers.
Or B) you think that there is something nefarious here and the API might associate who you are or where you are, and store or distribute that data.
If it's A) then you should be fine. If you think B) then you shouldn't use a phone from Apple or Google. Because as far as you are aware, they share your location and personal information.
As far as integrity goes, I can't see a situation where you would both accept running an iOS or Android phone but at the same time avoid apps with this API out of privacy concerns!