TDS meter works by using some carbon electrodes to measure the conductance of the water. Simple to do and calibrate, and no parts to wear out, the calibration is simple to do and the dimensions of the cell are easy to measure to calculate the output.
The nitrate and such kits use the water as part of a chemical cell that makes a small voltage that is proportional to concentration. You need a specific set of electrode materials, reagents that only respond to the specified chemicals that you are interested in, a temperature sensor to provide compensation and a micropore filter to allow the reagents to contact the water. All consumable materials that are both hard to make and use with the required purity at home easily.