Most chips have a capacitor across the power pins. That would tend to "integrate" the signal.
Recent chips operate at sub-volt values but the LED is in the 2+ volt range so they likely are on different rails of the power supply or have buck/boost circuits.
Even trivial designs don't use full in-order processing so multiple ADDs might overlap a MULT. This ain't your father's first processor.
The traces on the board, if not carefully designed, likely have reflection and ringing effects, especially at via points making the signal rather noisy.
Processors are running in the nano/pico range but the led/camera likely can't reach that resolution.
At these speeds the resistor-to-ground circuit on the LED would have capactive and inductive effects that would be hard to model and need to be pre-measured.
If it is a near-field (touch card) key then there are a lot of other noise sources including other chips and circuit trace induced voltages.
Cryptographic coprocessors in Verilog/VHDL are available as open source or vendor IP. They won't leak.
I could construct a "bench level test" that MIGHT be able to detect the computation but I'd need a really good oscilloscope on a well engineered circuit board knowing the exact specs of the processor and knowledge of the algorithm used.
Oh, and as mentioned, nobody who knows anything about crypto already knows and protects against timing/power attacks.