On a Witty Pi L3V7, with one battery but not another (and a good power source) the blue LED flashes quickly and constantly after finishing charging up. The web interface shows: Vout=5.13V, Iout=0.23A. The connected Raspberry Pi 4 is working OK. If I switch off the power source to the L3V7, the Raspberry Pi goes off as no power reaches it as the L3V7 has instantly cut the power. What is wrong?
The blue LED connects to the CHARGE pin of TP4056, and the red LED connects to the STANDBY pin.
The behavior of these two LEDs are only controlled by TP4056, and they do not affect other functionalities of the board.
According to your description, your battery may be faulty and it's actual capacity might be too small. Your power source on USB type C port worked as the only power source for the board: if you remove it, the board will not be powered.