It probably has something to do with bending the top of the mast when you pull down which probably gives the sail a little better shape (more curvature). In my case I am able to sail almost directly upwind by pulling the sail as tight as I can get it and pedal lightly. On light days I pass all the other sailers in their big boats. I think it has something to do with pedaling lightly, the sail is able to create an apparent wind of some sort pulling you along. If I close the sail and pedal only I go 1-2 mph, if I open the sail and pull it tight I go 3-4 mph while pedaling lightly vs 1-2 mph without pedaling in light winds (5-7 mph which are typical around here).
I've found this too. There seems to be a synergy between sailing and pedalling slowly when heading just off the wind in light winds. I actually recorded a GPS track of it on one occasion:
The wind was shifting slightly, so I needed to zigzag to maintain the optimal angle to the wind. As you report, I was able to maintain about 4 mph. I don't know what the physics of it is, but my theory is that the gentle pedalling is enough to negate the resistance of the hull through the water, so the boat becomes a virtual ice-boat and the lift from the sail is all profit.
I think the need to keep the sail pulled in tight is to maintain an optimal angle to the wind and is separate to kayakman's observation that pulling down on the clew enhances performance. There are times when I've found that pulling the clew down helps even when travelling downwind, so I guess, as you say, it's to do with improving the sail shape.
PS (Aussie perspective), in winds much over 25 mph, in an AI, the trick to travelling upwind is to furl the sail so there's only about 2-3 feet of it out, then to tack at a very broad angle to the wind (about 70 degrees).
Here's one of okz00k's tracks. The southern track was travelling upwind into 30 knot winds, with just a bit of sail out.
That was also in an AI. Don't know if it works in a TI.