What do I think?
First, "Ugh! Memories of 2e flooding back into brain. Must... Represss..."
Second, what you're talking about is NOT a minor change to the system. In order to work well you would have to completely retool the combat system from the ground up taking into account reach (as Lord Pendragon mentioned) etc. in order to get a small increase in realism.
To illustrate the effect of the change you're considering, let's take a look at your examples. Off the top of your head, you estimate it might be fair to give the dagger twice as many attacks as normal and the greatsword half as many. So, we come up with the dagger attacking four times for every time the greatsword attacks. On the face of it, this solution has the dagger effectively doing 4d4 damage in the same period the greatsword does 2d6. So a dagger averages 10 damage per time period and a greatsword 7. A longsword that attacks twice in that period (assuming it to be the baseline) would deal 9 points of damage. If you think it's realistic for everyone to bring a knife to a swordfight and for the dagger to be the most deadly weapon in the game, preferred to swords and spears, that may sound like a reasonable result. I don't think it does.
But wait a minute! That's not how D&D is played. By even mid levels, characters do more damage from strength, weapon specialization, magic, bardsong, etc. than from the weapon dice. So, let's see what happens if we take three 8th level fighters with 18 strengths (15 +2 for levels, +2 gauntlets of ogre power) and see how much damage they inflict per round. They've each got weapon specialization in their chosen weapon and a +1 shock weapon.
Mr. Dagger: 4 attacks for 1d4+7+1d6 each--52 damage per round
Mr. Longsword: 2 attacks for 1d8+7+1d6 each--28 damage per round
Mr. Greatsword: 1 attack for 2d6+9+1d6--19.5 damage per round
You'll notice that, once strength and magic is included, the dagger is, by far, the damage king, dealing 2.5 times the damage of a greatsword in one attack period and almost twice the damage of the longsword.
I submit that, unless you revamped the system entirely, there would only be one viable weapon in such a system: whichever one gave you the most attacks.