The circuits that you posted in post #19 would have worked. The schematic you just posted won't work.
The MCP73831T is designed to charge Li-Ion batteries at 4.2 V, so it needs more than 4.2 V to fully charge the battery. You have supplied it with 3.3 V so it won't even start to charge the battery.
The circuit with the BAT20JFILM and the Si2371EDS will never conduct current in either direction.
The circuit that you posted in post #19 would have worked fine. Even if you change the ICs, the idea of those circuits is good, and you can adapt the ICs that you are using to that.
The first picture of post #19 is a "Wired-OR", which takes either the USB supply via the diode, or the battery via the MOSFET, and supplies the +5V line (which will not be at 5 V, it will be a bit less, but that's fine)
The second picture of post #19 is the charging circuit to charge the battery from the USB supply.
The third picture of post #19 is the regulator that takes the +5V line and regulates it to 3.3 V.
If you want the same functions, you should replicate the blocks that are shown in post #19
I have drawn out the circuit suggested below.
View attachment 131967
I have added a 10k resistor to ground on the input. That is to make sure that the VBUS is very low when the USB is disconnected. Without that, the MOSFET might not turn on properly. The 10k resistor might not be needed, but without it the VBUS could be at about 3 V.
The 100 k resistor in series with the LED will mean that it is very dim.