How comes lift is less than weight on take-off?
Indeed, what puzzles a lot of people is that excess lift (excess as in more than we need for weight) it not what makes an aircraft climb. 

Of course, a take-off is a dynamic situation which doesn't really call for a balance of forces and momentum, so the situation is a bit more complicated to explain, but let's try a thought:

Imagine it this way: If lift > weight you will accelerate 'up-'wards (ie. balloon style). Air will come not only from the front moving backwards, but also from above moving below. The relative air moving past your wing is now coming from front and slightly above. This decreases your angle of attack and thus decreases lift, usually up to the point where lift = weight again.

If lift&lt; weight, the opposite happens (ie. accelerate downwards) and air will move from below the wing towards up. This increases the angle of attack, increases the lift, and again lift will match the weight. 

It's basically a statically stable system, where the only really stable position ,if you think about, is when lift equals weight (in straight-level flight, mind you. For climbing and descending you have the formula stated in your question).

If you want to climb, increasing lift is not the way to go !