Why does this SMD LED only work on AC?
There are some LEDs in that size package (2835) that work at 18V, such as the Cree J-series. I assume (and I could be wrong) that they are effectively a number of junctions in series on a single die, since the die count is specified as "one" on other manufacturer's datasheets. 

Perhaps there is a higher voltage version this manufacturer is using to make 230VAC lights simpler, like 36V, or (more likely, I think) they are using two 18V dies in the one package.

At the peak of 22VAC RMS (~31V) a 36V LED would certainly be emitting a fair bit of light, though not nearly at full brightness. 

Good question, that's an interesting evolution in LEDs. 

Edit: Here is a real-life example of a two-die 36V white LED that may be the exact one used in your lamp: