Yes. The NHS app requires virtually no data because all you need in reality is a phone number to send a text alert - that's it. You're right as far as this being an article of sensationalist journalism with no evidence presented. The reason though why people don't read Ts and Cs is that if you don't agree, you can't use the programme/app. The small print is usually too long, probably to deter people from reading them.
I too wouldn't want to use a pubco app, especially not the ones that ask for bank details and would give fake info when asked, save for a correct phone number and yes, it would be far simpler if everyone used the NHS app - I don't know why the Government doesn't make it compulsory and if it's a requirement for the hospitality industry, then why not shops?