I took my personal unit apart to make some pics to show you what may be your problem. Firstly no the 3ds will not boot if a battery is not present. The battery (even if dead) completes the circuit. If you plug it in without a battery a light will come on briefly, because it's checking itself. Once it discovers there's no battery the light will go off.
So here's the deal. This is fairly common in Nintendo handhelds. The contacts in the battery compartment are a little bit like springs. Over time, due to the constant pressure of the battery pressing up against them, they begin "sinking inwards". Eventually they "lose contact".
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If this is what your problem is there are two "fixes". The easier one is to wedge some paper or something (thick enough but not too thick) into the crack on the top edge of the battery.
[image|3208677]
This will push the battery downwards more and possibly restore connection. How long it will last, no telling.
The longer term fix is to pull the contacts back out a little. Using a pick tool of some sort. I made my own out of a "safety pin", the end of which you can see in the pictures. Using the hook tool insert its end into the contact, they look like 6 contacts but it's really 3 pairs. Hook both in a pair. Like so.....
[image|3208676]
...and very gently tug directly away, arrow indicates direction. They are somewhat fragile. Don't pull too hard. You only need to bend them back out enough to make consistent long term contact with the battery. See how they line up almost perfectly with that horizontal trace? That is about as far out as they need to go.
Again be gentle, go slow, make small changes. Try to get them all as even with each other as possible, however "perfect" isn't required. Test, if all is well great, if not tug them a pinch further, test again. Eventually you'll get the sweet spot. Good luck.