I, too, was taught the strange stab into the belly, which could then be used to rip out the spleen, kidney, or whatever else you might find in the gut. And for a long time, I never thought to ask what if the guy is wearing a leather jacket. And so, for years, we practiced the spearheand - always in the air, never against a target.
I do agree with Steve Weigand's explanation, there are grappling applications for the spearhand. I'm not sure the other hand's position is easily explained; however, that doesn't discount the grappling application.
So I will offer another explanation that does take the other hand into consideration.
Take a look at this video. In it, Chiba demonstrates iriminage, which is a throw by entering into your opponent's space. If you look real closely, you'll see Chiba, at :52 is throwing his opponent. Look at the "spearhand". And note where his other hand is: he's guiding his opponent, just like in the form!
Of course, a huge difference in this throw and the spearhand in most TKD forms that use a spearhand is that the thumb is up. But Chiba's thumb is down - a very important, but subtle difference. With the thumb down, that allows the arm to roll into the opponent. Keeping the thumb up, per the TKD form, allows the opponent to more easily roll out of the technique before the throw is applied - an escape or reversal. So to mitigate that, we throw with the thumb down.
But there are viable applications where the thumb can be up (and the other arm still in place per the many TKD spearhands in many of the forms).
Take a look at a completely different kind of throw here. In this throw, Yamada is performing a cross-throw, or X throw - jujinage. It is also an entering throw, but note that the thumb is upwards - and the other hand is used to effect the X in the throw.
More subtle observations:
When you issue a strike in TKD, note it's almost always via a fist. And half our blocks are done with a fist, and others are done with an open hand. What's up with that?
Well again, not all is as they appear. A "fist" has many representations. Of course, you can punch. But why do we block with a fist, and then not even use the fist as part of the block (eg, momtong makki, a middle block, where the "blocking" surface is just below the fist).
The answer is that many of our techniques - strikes and blocks - are grappling maneuvers. A fist usually is representative of grabbing something, whereas an open hand (knifehand, or sonnal makki) is not grabbing anything.
So the spearhand is not a fist - it's an open hand. And if you look at the two videos, you'll note there's no closed fist - nothing to grab!
You didn't ask, but there's more. Note the stances. Have you ever thrown a spearhand with the opposite foot forward? Probably not. There's a reason for that. In these two throws, you are inserting yourself into your opponent's space - especially the jujinage/x-throw. And when you do that, your opponent is on one side of you or the other.
In Chiba, at :53, note his right foot is forward, and he's using his right hand (thumb down). In Yamada, at about :10, you see his left foot forward and he's using his left hand for the primary throw. Also note these two throws are "entering" throws: the opponent is more or less in front, and is being turned back to the direction from whence they came. Although not a strict rule here, it's more of a guideline, but the stance to use for entering throws is always a front stance.
And a front stance, with its stability, and the foot alongside the opponent, allows to assist in a trip or sweep, and offers more stability for the thrower (the issuer of the spearhand). That's why spearhands are almost always thrown with the same foot forward: stability, control of the opponent, and to better effect a sweep or trip. Try doing these throws with the other foot forward, and you will feel much less stable, and an opponent on his way to the ground can more easily pull you down with him if he gets ahold of you or your clothing. This is much harder to do with same hand/same foot forward.
But wait! There's more!
Note that in Do-San, after you issue the spearhand, you do that funky turn with the hand behind your back. What's that all about?
That's a handshake throw (aka wrist throw... this has many names). IMHO, this is a parlour trick - something fun to show the kids or at parties. I've never really seen it used IRL, although all of the principals are there. But here is the throw. Note that you are engaged in the handshake, your turn is the throw. One thing that should be glaring at you is that in this video, defender is extending the same hand as you do in Do-San, but he's turning in the opposite direction. There are several variants to the handshake throw, and this is just one of them. This video shows a few others.