Water Leak

I had a leaking deck drain hose the moment I launched my brand new boat, the bilge pump started up almost instantly.  Luckily it was located where I could see it when I opened my I/O's engine cover, just tightened the hose clamp and was good to go.
So I think your assessment is spot on.  One way is to plug each drain fitting on the outside of the boat and see if the water stops.  If it does, its just a matter of removing plugs until the culprit reveals itself.  
You absolutely should have an automatic bilge,  You need to diagnose that system as well.  The bilge pump and switch get power direct from the battery.  Obviously the pump works (lucky for you), either the switch is not getting power or it is defective.  The switch can be either a float or electronic type.  Brown wire is used for bilge pump circuits.
Here's a simple bilge pump circuit diagram, the bilge switch and pump are wired directly to the battery.  Since the system is wired directly when the helm mounted bilge pump switch is "off" the automatic switch activates the pump when required, when you turn "on" the bilge pump switch, you bypass the automatic switch and turn the pump on:

This is the bilge pump/switch that is in my boat.  The switch is an electronic one:


Here's what a typical "float" type switch looks like:

Here's the ABYC Wire Color Chart to help you identify wiring:

The bad news is that most of the drain hoses are installed while the boat is still split apart.  The fittings have the hose flange on the interior secured with a hose clamp.  The fitting is secured to the hull using a nut.  The fitting cannot be screwed out of the outer hull side  to the best of my knowledge.  Here's what it may look like:

I got lucky that the fitting I had trouble with was reachable.  Hurricanes with outboards will have a much tougher time reaching them, if at all.  One method you can potentially use without having to split the hull and deck is to cut a hole in the deck over or close to the drain location, address the problem, then install a deck plate in the hole to cover it up (usually deck plates would be around 5 inches in diameter.