Why does TWINKLE use light instead of current? TWINKLE is a device devised by Adi Shamir to optimize the sieving step of GNFS. It consists of a cylinder, at the bottom of which are LEDs corresponding to factor base primes which blink with frequency 1/p. If more than a certain number of the LEDs blink at the same clock tick, you've found a GNFS relation (smooth integer pair). This is measured by a brightness sensor at the top of the cylinder: if the light level is above some threshold, the necessary number of LEDs must be lit.

My question is: why use LEDs instead of just measuring current? You could replace the LEDs with transistors that alternate current instead of blinking, and if the current sums to a certain level you have a hit. Why wouldn't this worK?
TWINKLE  uses LEDs as their output is easy to sum.  Simply put, the overall light level measured by a photo detector in the top of the tube is the (rough - important) sum of the individual LED emissions. I say important as the overall sum does not have to be accurate. It only has to exceed a threshold level when TWINKLE notifies the controlling PC of the current integer. This ambiguity mitigates   the practical issues of light travel distances /times, variations in individual LED emission levels and geometric imperfections of the photo detector's lens.

It's really quite clever.  This guy's going to do well for himself.  You can add up the output of the device's 100,000 processing cells with a single detector that has clear sight of all of the emitters. The innovation here is that each cell can operate totally independently and an overall 100,000 input   adder is not required.  It also allows the outputs in the form of light to be send at 10GHz, whereas that's getting ridiculously fast for circa 1999 technology distributed across a 6" wafer.

Your suggestion of adding the currents /measuring the total resistance  would require something like this arrangement:-



A long chain of 100,000 resistors would have to run between all of the computation cells. To make the maths easy for a sanity check, assume that the wafer is actually square with a grid of 316 x 316 cells. That makes the resistor chain approximately 42m long.  The reasons  that this can't possibly work are entirely off topic for this site, so here they are:-


You cannot guarantee that any circuit produced on a wafer is viable. In 1999 manufacturers would have been using a ~200nm lithographic process to make this stuff.  Adi himself suggests a device yield of 80%.  That means it's highly improbable that the resistor chain would be intact without any break or significant thinning. Surprisingly, the yield rate is dropping as technology improves(?) and may be down to 30% today.
It's impossible to measure resistance. You can only detect a voltage or voltage driven current. And any changes in current or voltage have consequences especially if the rate of change is fast.
Contrary to what people think, unfortunately there is no such thing as digital electronics.  There is only fast analogue electronics. The implication of this is that electricity can't even travel at light speed, and can be considerably slower (50%c - 80%c). The resistor chain couldn't be longer than ~5mm for transmission line effects to start clobbering the signal at 10GHz.  
The same pesky transmission line effects make it impossible to synchronise all the cells to within 10ps - 20ps that Adi requires. Hence the light sensor on each cell.


It's worth noting that TWINKLE's successor, TWIRL, does not use opto summation techniques.  This may be an indication of the designer's lack of confidence in the benefits of optical /analogue methods for modern computing