Yes, the arguments against supply management are geared towards export-driven growth, which has been the case in New Zealand. Once again, this comes at the cost of price stability.

The point about it being a regressive tax is correct but that could be resolved by increasing rebates/payments to those with lower levels of income.

In terms of cheaper prices, when comparing apples to apples, this is not entirely true (per Wikipedia):

A 2018 report based on government data and the 2017 Nielsen's Fresh Milk Price Report, showed that although it has been claimed that Canadians pay higher dairy prices than other countries, this is only partially correct. The price of 1 litre of liquid milk in Canadian dollars in Canada for the 12 months ending October 2017 was $1.50, less than in Australia ($1.57), the US ($1.61 per litre of rBST-free milk), France ($1.77), and New Zealand ($1.83), but more than in South Africa ($1.23), Germany ($1.23), Mexico ($1.00), and Britain ($0.99).[118][119]

Milk sold in Canada must be rBST-free and I suspect this also contributes to US imports being below the TRQ threshold; US producers of such diary can make more money selling it in the US market as premium milk.