Problem with that approach is that, if true, it can only really go in "a bit".. i.e. A few mm, which won't give the screws enough purchase as the framework of the door will just split again (as you'd expect if you drove a screw into a bit of wood a few mm from the edge.)
The quickest and dirtiest way to repair this door to functional would be to smear that crack with wood glue then blow it into the split, repeat, then drive 4 screws through the face of the door so there's a screw either side of each hinge screw hole, clamping the split back together, wait a couple of hours for the glue to set, and wind the hinge screws back in again. If the aesthetic of screws in the door face is a concern, filler and paint will sort it. Tbh, I'd just paint them