We will have to disagree in this inst..
Again, I will raise an air of caution:
There are differing degrees of mastitis......most and that is probably 80% of cases can be soothed with a hot flannel and expressed, some go beyond that and some dams become extremely poorly, and the teat not only turns into a solid jaffa orange and splits, but as in my case (hence being more cautious than many of you) also turned gangrene, my bitch losing her teat completely,
That was to my determent in ignoring my vet and following everything I knew of mastitis, listening to my girls breeder, and everyone else in the dog world, I continued to feed my pups on unaffected teats, continued to soothe and express the infected one, I should have allowed the milk to dry up with the pills, as my vet wanted, and my girl would not have lost a teat..............
We don't always know best.........sometimes........I know it is rare especially when it comes to breeding,

but just sometimes, our vets get it right.
If the dam is in distress, if she is ill, and if the milk is drying up anyway, why continue to allow milk to be made if the teat is in trouble? The vet is the one who has seen it, we haven't, and we can't see just how ill the dam is, as the pups are old enough for early weaning, then IMO they need weaning as the vet has said, my vet cared about my bitch and her welfare, and I agree that a dam comes first, she's stressed and in pain.
As I said, it is more work for a breeder, but I would rather have a happy dam......... and there are no worries that the pups will not survive, the smaller one I would keep on the syringe until 3 weeks, and hopefully by then she will be fine to lap with the others.
And the dam will have a full recovery, fingers crossed.

I just know how terribly wrong things can go, and how mastitis can differ from one case to another, I really would follow the vet here.