I've tried to build the oximeter from scratch, with a LTR-301 photo transistor and it doesn't work as expected.

The problem is the LTR-301 spectral response which is quite narrow around 940nm. So it doesn't receive the red light!

To build from scratch (without hacking a KY-039) we need to use a wide range photodiode, such as the LPT 80A which can receive both red (600 nm) and infrared (940 nm) light.

Maybe finding a phototransistor with Srel (Relative Spectral Sensitivity) value around 600nm very similar to the 940nm (in this one it's 50%) is also better: