which is the advantage using the modulated path with the LDR in one leg and a fixed resistor in the other rather than the more usually seen connection of the LDR between signal and ground standing alone ?
This is an add-on tremolo unit, apparently with the goal of maintaining a signal about the same size as the original dry signal.
Look at the 12AX7 between the input and output. The effective plate load for a.c. signals is the parallel combination of the 47kΩ plate load and the 100kΩ & 15kΩ resistors in series. The other 100kΩ and LDR are isolated by the 1MΩ and the LDR's own high off-resistance. So the effective plate load is ~33kΩ.
Given this low effective plate load, the gain of the 12AX7 is 34.8 with a fully bypassed cathode resistor, or more like 17.5 with no cathode bypass cap.
The 100kΩ/15kΩ series resistors to the right of the 1MΩ pot form a voltage divider with a "gain" of 0.13, which is to say it knocks the 12AX7 output down to ~1/8th of what it was initially. With no bypass cap, the dry output is 17.5/8 = ~2 times its original size, which in voltage gain stages is "no-gain."
The LDR has a similar voltage divider setup, so you get smooth variation from full-trem to no trem without changing the size of the original signal.