Lithium ion battery protection IC not able to drive MOSFET
I think I found the solution to the issue, and also posted it over on the eevblog forum: https://www.eevblog.com/forum/projects/pain-and-suffering-getting-the-dw01-and-8205a-protection-circuit-work-right/
The trick is to use a slightly lower value for the capacitor marked C2 in the circuit you posted. The Chinese boards seem to do the same. Use a value like 80nF instead of the 100nF mentioned in the datasheet, this should cause the circuit to switch on when battery is connected without the need of a pushbutton.