the LNK chip seems to be exactly the sort of thing you're hoping that the LED drivers would supply. A small and only marginally regulated replacement for "ye old" 60Hz transformer and DC rectification circuit, that converts AC line voltage to something appropriate for the input to Arduino-like circuits, in a hopefully cheap circuit. That's what it says in the datasheet:
LinkSwitch-LP switcher ICs cost effectively replace all unregulated isolated linear transformer based (50/60 Hz) power supplies up to 3 W output power.
dhenry is also correct that it's a gated oscillator circuit. It says THAT in the datasheet too:
Unlike conventional PWM (pulse width modulation) controllers, it uses a simple ON/OFF control to regulate the output voltage.
With the suggested circuit having 220uF of output capacitance, and probably running that gated oscillator "as needed"
A lot of the confusion seems to surround the use and mis-use of the term "constant current" WRT power supplies.
A proper LED driver should a TRUE constant-current supply, holding the current constant by varying the output voltage "quite a bit", in order to accommodate at least variations in Vf of the LED string, and perhaps a relatively wide range of number of LEDs in series. You can see this on chips specifically designed to be LED drivers; for example the Allegro Micro LC5220 series supports Vled of 6 to 90V in its "buck mode" configuration. (Datasheet) Meanwhile the "CCCV" supplies are more "Constant voltage first", with some current limiting capability, and "well defined behavior" as the current increases beyond the limit (unlike, say, a 7805-style regulator.) I don't know whether these chips are designed to be operated in their constant-current "operating region"; it looks more like the CC feature is a protection mechanism. It might take a power-supply engineer to figure it out
So... Does a random "LED Driver Module" do anything useful, WRT powering hobby electronics? The fact that they're advertised as "3 x 1W driver" or similar implies a pretty low output voltage range, so they're probably not "true" constant-current drivers. And there's no sign that any two supplies operate similarly, or use the same circuits. You might get lucky, or you might not, and I don't think there is any way to tell without either getting much more detailed specifications than tend to be listed on eBay, or actually buying one and analyzing the circuit and/or behavior.
Given the average level of Power-supply-design expertise of Arduino forum readers ("nil"), I find the suggestion that they might be able to use some random cheap LED-driver module from eBay as a power supply to be ... dangerous. That's probably the reason that tempers in this thread are running a bit hot.