There are a few topics related to these issues but I didn't see any resolutions.
"Board not being powered by battery" - all of my batteries arrived with the polarity reversed from what the Arduino wants. Make sure the positive pin is on the left when looking at the top of the board from the connector side.
"My battery won't fully charge" - after using the code found in this tutorial MKR WiFi 1010 Battery Application Note | Arduino Documentation I also could not get any of my three batteries to fully charge, or so I thought. My Fluke 175 showed 4.2v on all of them. I bought a second MKR WiFi 1010, same results. I started "doing the math" line by line in the above article and found they were using INT for a variable with a calculated value of 4.389 which was being shortened to 4.
With the original code and the max_Source_voltage shortened to 4 it reports my battery at 3.91v.
With the updated code and the max_Source_voltage now set to FLOAT and 4.389 it reports my battery at 4.29 which is much closer to the actual 4.126