Do small, modern LED lightbulbs require external heatsinks?
Edit: Now you've posted more pictures, I see what you're doing.   Let's take a look at your presumptions here: 


Anything that makes light must surely have a "bulb" of some kind. 
The bulb is fragile by nature, and it will inevitably burn out. 
When a lamp fails, it's almost always the bulb burning out.
Scrapping a fixture when a bulb burns out is insane; it must be replaceable somehow.  
If a manufacturer says it's not replaceable, that can't possibly be true. 


All those presumptions are dead wrong with LEDs. 

That is a packaged consumer product with a permanently mounted LED in it.  It contains several components, including packaging, a power supply which converts AC into the constant-current the LEDs need, a beefy heat sink, the LED emitters proper, and lensing to direct the light.  

Yes, you need heatsinks

LED emitters used for illumination make a lot of heat.  Some is from the LED itself, and that is decided by how hard the LED is driven.  Some is from the inherently inefficient phosphors which change blue light into spectral light. 

LEDs are much more efficient than the old incandescents, but even so, almost all the watts of energy put into an LED emitter turns into heat, not light.  That means the heat sink must be rated for the wattage of the LED.  The LED wants to be kept as cool as possible, and the cooler it is kept, the better it will run.  But you're still going to have a lot of heat. 

There is basically no way an LED of any brightness is going to function for long without a heat sink.

So why do they fail?

Properly cooled, an LED emtter is likely to outlive all of us.  If there's anywhere cheap builders will cut costs, it'll be in the power supply.  This is the part which inputs 120/230V mains voltage and outputs a constant-current DC for the LED emitters proper.  Capacitors are the most sensitive component - they ordinarily have a finite service life, and you have to go out of your way to spec the really good ones or spend a bit more on a capacitorless design.  The top failure points are: 


Power supply (particularly capacitors)
Lamp wiring
Environment damage
LED coming unglued from the heat sink, causing overheat
Blockage of airflow to the heat sink 
Lead-free RoHS solder crystallizing over years - reflow the solder or use lead solder 
Lenses fogging up or discoloring
Cheaper LED emitters getting dim as they age 


The LED emitter is rather unlikely to fail.  What does this mean for fixture manufacturers?  It means making LEDs removable is a waste of time and actually makes the product less reliable, because a socket problem is more likely than an LED emitter problem.  

With the very concept of a "bulb" gone, it gives the manufacturer freedom to design fixtures as they please.  

This means you were barking up the wrong tree when you took it apart trying to find the mystery replaceable bulb.  There was probably nothing wrong with the emitters proper, unless they came unglued from the heat sink.  Given the age, most likely it was a power supply problem.  You could have tested for this with a voltmeter. 

More about overdriving LEDs

LED emitters (the chip inside the "bulb" product) are expensive, a common cost-cutting approach is to overdrive LEDs.  The manufacturers fully expect this and spec the LEDs for this.  For instance this Cree XP-G2 is rated at 1500ma but is specced at 350ma.  Notice most of the charts assume 350ma, or assume 85C. 

This unit (in R4 code, nominal 130lm) produces at 85C (and you can follow along on page 11, electrical characteristics and relative flux vs. current charts):


At 350ma, voltage is 2.8, watts are 0.98, lumens are 100% or 130, lm/w is 133. 
At 750ma, voltage is 2.93, watts are 2.20, lumens are 195% or 253.5, lm/w is 115. 
At 1125ma, voltage is 3.05, watts are 3.43, lumens are 270% or 351, lm/w is 102. 
At 1500ma, voltage is 3.14, watts are 4.71, lumens are 340% or 442, lm/w is 94. 


This assumes 85C.  Go to page 10 and you can see exactly how cooler temperatures result in more light; 10% more at 25C, and 10% less at 125C.  Now look at thermal design on page 13 and you'll see where the device requires a derate for high currents at high temperatures. 

Lamp manufacturers balance the cost of more emitters (to run more efficiently) versus bigger heatsinks on fewer emitters.  Generally they need a heatsink regardless, so making a somewhat bigger heat sink is the cheapest option.