Number of parts is not really indicative of anything here... you don't need many as most of it has been integrated into an IC.
Odd that there is practically no mention of the "Tesla" brand or that part number anywhere else on the Internet, although TI has a Qi charging IC series they call bqTESLA. Would be interesting to RE that, but might be hard since it looks like 4-layer PCB.
And a bq51013B is a receiver...
I know the number of parts isn't directly indicative of much, but there are some primary requirements.
This is what i think i see on that board;
a tl431 regulating the power to that tesla chip or providing a reference.
the tesla chip, driving an led and driving 4 transistors
4x transistors, driving the tank circuit
a lm324 opamp, probably with a shunt resistor to stop the thing cooking if a metal object is placed on it directly. IF *IF* is actual wireless charging chip, also probably driver for coil communication.
Nah, none of the TI wireless transmitter products come in a package like that. I had a pretty thorough search for that chip too, all of the tesla related results returned refer to and old ic manufacturer, and i couldn't find a single wireless tx chip for this application, in that package.
Also, yes, i know the bq51013b is a receiver, it was on a flex pcb with a pcb antenna for retrofitting into an existing device or something. I got this in addition to my charger for free.