So, can this monitor heart rate — entirely without smartphone apps or proprietary desktop apps — using “Every Breath You Take” and other open-source apps?
I think that link is referring to going for a workout with just the monitor itself to record your heart rate - no other devices present. You can use/build your own mobile or desktop apps without needing their software if that's what you're wondering.
I made a custom iOS app for my wife and I to monitor our workouts using their SDK and I really happy with it
That article is referring to the internal memory that the Polar H10 has, which can be used to store some data. But you will eventually need to connect the device to retrieve that data. You can access this data programatically with a blue-tooth-low-energy client like my application is doing (using python)
EDIT: Seems like a “no”: https://support.polar.com/en/support/can_i_record_my_heart_r...