I personally wouldn't mind if my (possibly randomized) BT address was made public, potentially identifying me as currently COVID-19 infected - heck, we will probably all have been on that list in a year or two as things stand today.
What someone (not me, of course) would probably mind, however, is if the geolocation data was used instead - and it showed what clubs I went to during the last 14 days.
And my point remains that, from a privacy perspective, that distinction is artificial and misleading. Your phone in this mode is seeing BT/BTLE devices all the time, which is why this (or WiFi / GPS profiling) work. On my route to work I pass by dozens if not more BT devices with static location, somebody with detailed log data over a few days will be able to mark both locations in a pretty fine grained manner. The proposed process is not and can not be anonymized sufficiently. Correlation is cheap these days, with big enough adoption a company with enough incentive or something like wigle.net for bluetooth devices will appear and the data will be misused. That might not present a viable problem in the short term during the current outbreak but the long term implications of tracking like this need to be discussed before releasing such a tool to the public.
There's other issues, e.g. a more direct and immediate scenario where Alice knows Bobs bluetooth MAC, as well as an infected in likely proximity, and dumps fake data into your service in order to quarantine him.
However, from your description of the Alice/Bob part, I'm still confused whether you realize that nobody is (or can be) quarantined in the proposed scheme?
The case you describe would, as I understand it be: "Bob is then advised by his own app not to visit his parents and be careful about not infecting others". A bad joke, sure, but not something that can "send him into quarantine", since nobody on the server side knows that he is in any danger.
I may be missing your point here - but if I am, please clarify and bare with me;-)