Yes, the headline is basically a straw-man. I am not aware of anyone who has made that argument, but I'm only using it to (a) draw attention and (b) make a point: the stock market "rally" is, if anything, confirmation of the basic story about the liquidity trap.
According to basic microeconomic theory, the price of a good is expected to be positively correlated with a change in the price of its substitute. If a gala apple and a jazz apple are each $1 apiece (equilibrium) and the price of jazz apples go down, one would expect for the consumption of gala apples to decrease and the consumption of jazz apples to increase, such that a new equilibrium price of each would result. Thinking of stocks and bonds as substitutes (in that each are claims on future revenue streams), one could conclude that it is only natural that as yields in the bond market decline (due to monetary policy and general pessimism about the economy), investors would turn to stocks in search of yields.
Here are some facts about the current market rallies:
(1) the S&P and DOWJIA are booming:
(2) Bond yields are down (bond prices are up):
(3) Gold is up:
And, obviously, the short-term treasury interest rates and the effective federal funds rates are up against the zero lower bound, which means the bond prices for treasuries are also high.
I ask simply: are these the signs of a recovered economy when unemployment is still really high and there is a large persistent output gap?
The answer is an emphatic NO! Clearly what is happening is that the demand for investment opportunities is outstripping supply. The real rate of interest that currently prevails is too high to restore full employment, despite the efforts at Quantitative Easing.
The conventional mechanism to bring about full employment in such conditions is for the central bank to conduct open market operations to decrease the nominal interest rate, but when short term rates are hovering around zero, there's nowhere to go but up. The two remaining ways to bring about recovery are to undertake fiscal stimulus, which would increase the supply of investment opportunities (because of an increased deficit financing the extra spending) and, by some mechanism, triggering inflation expectations, which would lower the real interest rate, even at the zero lower bound.