There is a free Building Footprint layer that you could use as a start.
https://github.com/Microsoft/USBuildingFootprints
From here you would download building footprints into a zip file for the entire US. (The data is not perfect as it is dated with missing structures as well as structures that have been modified and or demolished and this would vary from one area to the next.)
You would unzip this file and then add the data to Pro (I think ArcMap has a hard time processing all the data).
Then use your jurisdiction's boundary to only clip out building footprints for your jurisdiction which would leave you with building footprints with no addresses. Since you do not already have building footprints I guess you would be using aerial photos to determine building location, but without the vector building footprint polygon layer I'm not sure how you would automate the process.
You can then add a new address field(s) to the new building footprint layer and then spatially join to the parcel layer and field calculate the address into the building footprint layer. Then you could create a point at the centroid of the building footprint. Then spatially join the address point to the building footprint centroid point that provides a corrected address point location with an address. If you already have address data for the address points, you would need to compare this address data to the parcel based address data as they may be different and you'll need to determine which address is correct. This will not get addresses for all points (As you say some address points fall outside parcel boundary which you can manually fix, but it should hopefully be a small subset of the original data).
