Detect the change in the R-L time constant:
Using LVC serie single gate digital ICs you can detect very small time constant changes. I have two pulses with 2ns time difference and detect when shorter one became longer than the other (D latch detects if clock or D input was faster).
For me your schematic is not clear.
1. What is battery shield?
Do it have several batteries so each 2 pin set are isolated from each other pair or it is simply several ways to connect to the same battery - in other words: do the pins 10 and 12 are internally shorted or not?
In the past I have built my own but to keep things simple unless I need a lot of amps I use 18650 power banks that I solder a pair of wires to like these for just 5v https://www.aliexpress.com/item/1005005883705502.html
or these if I need both 5v and 3.3v:
ttps://www.aliexpress.com/item/1005001946760259.html
You showed battery containers for single cell, but at your schematic battery shield have many outputs. I still don't know if pins 10 and 12 are internally (in battery shield at your schematic) connected or not. If these are separate voltage sources than they would be separate at schematic. If they are shown as one source then what it is intended to tell us. Is it one voltage source with just many outputs connectors (so pins I'm asking are connected) or separate batteries (so pins are not connected).
It is important to know it. If transistor emitter is the only connection point between two in other way galvanically separated circuits than it can't provide any information between them as current always needs 2 wires (I hope you know it).
So until you don't know if esp32 have the common supply than the rest of circuits it is hard to understand what circuit author had in mind.
I would like the module/ESP32 to take an input from the TDA0161.
Was your intention to use transistor as voltage follower (input at base, output at emitter (about 0.7V lower)? But you connected transistor collector to - and not + so it is not clear if it was your intention.
You connected transistor in reverse polarization mode (base collector junction forward polarized). This mode is sometimes used in analog circuits as it allows to get very small voltage drop between collector and emitter provided that driving current is higher than switched current. For esp32 input it is not important if when transistor is switched on you have 0.1V or 0.01V at it so there are no logical arguments for such using this transistor. In this connection you loose transistor current amplification from may be 400 down to may be 5. But as you need no amplification at all than I don't know what was your idea.
In both cases everything depends if two voltage sources get from battery shield are connected or not and I still don't know it.
From what I saw on a video ... of the circuit the LED should display when metal is detected by the TDA0161.
See at your schematic. Your LED + its resistor are permanently connected between + and - from battery. TDA0161 have no influence on this circuit.
In your case I would start from testing how the ring you plan to use influence on L.
L is mostly changed by ferromagnetic material in close proximity to it.
Shorted coil changes mainly inductance Q factor (representing power looses in inductance) and less its inductance (L value).
What about dropping there TV pilot and having somewhere IR receiver just detecting that pilot was used.
Or a whistle and a sound receiver.
Being in secondary school I have made circuit to switch on/off my desk lamp by clapping.
In my country we have here one day in year when it is customary to douse oneself with water in the morning (a much reduced old folk tradition). When I was 8..10 I have build from two pieces of sheet metal a switch that was opened when a bottle of water (next to my bed to be used in the morning) was standing on it. Connector was connected to battery and buzzer I had from 'Morse-electric' toy.