Nope. Not buying the “good intention”.

Are we going to do the same for all who ride public transportation, visit stores and any other of a number of things where the virus could be transmitted?

The “fact check” article says that first of all, this is not a requirement, but a recommendation that event organizers keep a list of attendees.
Second of all, it is not specific to churches - the article notes that the recommendation applied to all businesses, such as retail stores, nail salons etc.
Obviously it’s going to be pretty easy for some businesses to put it into practice, such as nail and hair salons where they keep an appointment log, and harder for others, like retail stores where people don’t give their name and phone number just to walk in and buy.
It’s a tossup as to whether a church will be able to do this or not. The Portland, Oregon plan was requiring people to sign up to attend Mass, so they would have a log. If Kansas City decides to do Masses differently, for instance first come first serve with no names given, or drive in Masses, they won’t have a log.
In any event I don’t care if the government knows I went to church. I’ll freely tell anybody who asks I went to church. If the government wants to come and persecute me for going to church, guess it’s martyrdom for me.