As I understand it, "organic" is a process standard, not a product standard.
The cows have to be raised according to the organic rules (Soil Association, in the UK). These rules may, or may not, lead to a product that is quantifiably different to that from conventionally raised cows.
If the quantifiable difference of interest is the omega 3 content, and a higher content makes milk 'healthier', there may be a higher probability that organic milk will have the higher omega 3 content and be the 'healthier' product. I say that because organic cows may have a higher probability of being fodder or grass fed, and this may result in higher omega 3.
However: IIRC, cows can also be organically raised on organic grain, and the milk that is produced thereby will be equivalent to the conventional raised cow milk in omega 3 content. If grain feeding cows is the more efficient and economic means of raising cows, and it can be done organically, then the supposed benefit for organic milk may be lost.
Sensible solution: use a product standard instead. Set thresholds of cell counts, B12, omega 3. If the milk passes, put a premium price on it. It may well be that grass fed, zero-hormone, free range cows produce milk that best fits that standard. But such cows are not always organic. The organic label is not a guarantee at present that the milk is of a better quality. Depends where the milk is sourced. And of course some conventional operations will produce high quality milk that also fits the product standard.
Pay for the product, not the process... And just possibly, the better product requires the more 'organic' process anyway.
(Apologies for any inadvertent strawmen in the above. They don’t seem to be too popular in this thread…)