Creating an electronic switch for controlling a headset's control button
Summary:  


As shown Q1 is drawing many amps of base current and the NOT expected 1+V Vce is a sign that the transistor is trying to glow white hot.
Add a sensible base resistor, say 3k to 10k, and it will work well in practice. (This is a much higher level of base drive than wold be used in most cases but will allow and extra low Vce on voltage. See below for details. 




A very major problem is that you    


haveused an incorrect simulation, which you have stated is incorrect,   
have achieved a result that does not present what you would get in reality,   
and have then concluded that this is what would happen
when you use a different real world circuit than what you have shown.  


Simulators such as SPICE can be excellent tools but you have to simulate the circuit that you intend to use as closely as possible, and you have to ensure that you do things which cause very gross departures from reality.

Consider - in your emulation, what is the base current of Q1?
 Adding a measurement of this value should be instructive, at least. 

With no base resistor the base current of Q1 will probably be in the many amps range. Using a [**TO3 metal can 2N3055""] (http://www.st.com/internet/com/TECHNICAL_RESOURCES/TECHNICAL_LITERATURE/DATASHEET/CD00000895.pdf) as an example which may approach this in real life you find that at Vbe = 1.8V, Ib = 4A, so at 3.6V it would be glowing quite nicely. 

Adding a base resistor to Q1 should restore reality.

You say "and expected drop of 1.026V in the CE junction" BUT this should read "a wholly and completely unexpected drop in the CE junction".  

By driving the transistor in a more normal manner here is what can be expected.
 Below are graphs of the Vcesat = collector base junction voltage for a BC337 - datasheet here.
 As base current is increased to about 10% of collector current Vce drops to around 0.05V for collector currents up to about 100 mA. At the &lt; 0.1 mA required in this case, using a base current of 1 mA or so, so that Ib >> Ic, will result in extremely low saturation (= on) voltages. 



The required voltages listed in the article that you referenced are >> 0.05V (see table below) except for "play/pause" which will tolerate somewhat more than the 4 mVshown so you should have no problem with Vce levels in practice. 



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