Milk is tested frequently and antibiotics are very rarely found [1]. Milk tankers that test positive for antibiotics are discarded and this milk never enters into the human consumption stream.
1. https://www.fda.gov/federal-state-local-tribal-and-territori...
>While the administration of growth hormones known as BST or rBGH to dairy cows is allowed in the US, it is illegal in Canada and therefore not permitted for use with any dairy cows.
https://www.healthcastle.com/what-you-should-know-about-milk...
>In both Canada and the United States, all milk sold must be antibiotic-free, so the issue is more about potential antibiotic-resistance than actually consuming antibiotics through milk. In Canada, a farmer who provides a dairy with milk containing antibiotics will have to pay for all expenses related to the milk that is thrown out. That means farmers take their commitment to providing antibiotic-free milk seriously.
Source - grew up on a NZ dairy farm. Every milk collection from every farm was sampled and tested for bacteria, and antibiotics. Too much of the first, and you were penalised with a lower payment for a start. Any of the latter, and you paid for disposal of all the milk in the tanker load, from your farm, and the others it collected. This was thirty years ago, if anything it will be stricter now.
Solution was simple, look after the cattle, and keep the medicated ones out of the milking herd.
Also the websites only talk about maintaining a database over this not that you need to discard the milk if positive.
Nobody wants out of spec milk because it's not useful. You can't sell it because it will come back to bite you.