Door leaking from the top?

I need the collective brain…

Our front door is leaking…from the top. That is, we have a drip (or rivulet the past few days) of water that forms at the top center of the door and runs down into the house. It seems to form at the top of the wood frame inside the door and then drop onto the door itself and eventually either down the front of the door or the back of the door (inside).

We’ve consulted one handyman, and his diagnosis is that it’s because we don’t have an overhang above our front door (not a full awning, just a small overhang). As such the wood above our door has gotten wet for a long time (i.e., years) and is beginning to rot…and with as much rain as we’ve gotten the past few days, it’s acting like a sponge. My concern with this diagnosis is that…well, it’s not like we took the overhang off. There never was one. Why would any builder build a home without an overhang if this is the result? Wouldn’t everyone in our sub be having the same problem?

So…for the collective brain:

1) Does your front door have an overhang?
2) Have you ever heard of this problem?
3) What do you think the cause is?