If you are worried about calcium carbonate clouding, then testing for Total Alkainity, pH and Calcium Hardness (CH) and Cyanuric Acid (CYA) will let you calculate the saturation index. All the minerals you cite such as magnesium, potassium, chloride and nitrates do not cloud water. Very high phosphates with high calcium can cloud. High phosphate and nitrate levels can have algae grow faster and that will initially look like dull and then cloudy water before turning green, BUT a sufficient Free Chlorine (FC) level relative to the Cyanuric Acid (CYA) level will kill algae faster than it can grow regardless of nutrient (phosphate, nitrate) level. My pool has had more than 3000 ppb phosphates yet is kept algae-free by chlorinating liquid alone (it's shown
here and
here).
TDS is mostly salt -- the calcium and bicarbonate from initial startup plus the sodium and chloride from chlorine usage since chlorine turns into chloride when it is used up and sodium comes from either bleach or chlorinating liquid or if using stabilized chlorine, pH and TA adjustment products (if using Cal-Hypo, the CH will rise). If one is using stabilized chlorine, then the CYA can build up and that lowers the active chlorine level unless the FC is raised proportionately so in that sense a higher TDS is a proxy for the high CYA, but it's the high CYA that is the problem, not the TDS itself. The pool industry has historically blamed TDS because they didn't want to point to the real culprit in most circumstances which is the buildup of CYA from stabilized chlorine and its reducing chlorine effectiveness against killing algae.
The conductivity tests used to measure TDS don't even measure the uncharged molecules in the water such as urea or chlorourea. Yes, there can be organics in the water, but they don't always show up as TDS when measured via conductivity. Even so, the amount of ogranics will be swamped by the amount of calcium, bicarbonate, and sodium chloride salt so won't be obvious anyway.
I write more about how TDS is sometimes used as a proxy for the age of the water in
this thread on improving the CPO course/manual.