Longer door hinge screws don't have enough reach on the house side, now what?

[RAMd®d]

If you use the toothpick method, drill the pilot hold to the proper depth, then put the toothpick in to shim the screw. If the hole is true, the screw will follow. It might be the tiniest bit cockeyed, but if the head fits the countersunk portion, error, if at all, should be minimal.

If you don't let the dowel and glue dry properly, it may spin if you drill with too much speed and push too hard.

But let it set up, drill the pilot hole with light pressure, and you should have no problem when putting the screw in the dowel.