Hi,
Not sure if this is applicable to your particular model but with some other Brother printers that have the same problem of not starting but can be started after performing a power refresh (the procedure that you're doing), it is due to a faulty capacitor on the main pcb.
This capacitor acts in a similar way as does a cmos battery in a PC in that it maintains the firmware settings in the printer from being corrupted for whatever reason which may prevent the printer from starting normally.
I think that it is the capacitor that I've highlighted in an image of the main pcb (part #LT3060001) from your model (according to the parts list - see p.31 ref #1 LT3060001 MAIN PCB ASSY MFC-L8650CDW) Your main pcb may have a different part number, but this was the only one that I could find an image of, but the others would be similar I think
I don't know what the specifications are for the capacitor but they should be printed on the capacitor itself. Be aware that if you do decide to replace it, take note of the capacitor orientation as it is polarity dependent so make sure that you check the polarity markings on the board and the capacitor i.e. +ve and -ve, to ensure that it is installed correctly.
On p.30 in the parts list it shows where the board is located but I can't find a service manual that describes how to disassemble the printer to gain access to the board.
(click on image).