I have soldered the wires from a brand new 500 mAh LiPo battery to the charger to make a first serious test. The internal resistance and high SoC put the charger in a constant voltage mode in about 10 minutes. I have measured the voltage across the battery (4.192V) to verify that it is not a thermal throttling.
The reason for using USB-A to USB-C cable is because I don’t have a USB power meter with USB-C connector. It is more handy to use USB power meter rather than measure the battery current with the multimeter. However, I have tested it with USB-C to USB-C cable as well, and it seems to work, since the charger gets warm.
The next step is to make a simple enclosure, which integrates charger with the battery.