From my experience discussing EVs with people, the biggest issue with per minute charging is that it complicates the question of "how much will it cost to get 500 kms of range". It adds another step to the equation, which confuses people. Imagine how much more difficult it would be to figure out how much it would cost to fill up at a gas station if gas was sold per minute at the pump, and each gas station (and even different pumps at the same station) had pumps that fill at different speeds. It would be much more difficult for gas stations to advertise their prices to people driving by, and people would have to do math as they're driving down the street to try and compare prices.
For the general public who goes out and is looking to buy an electric vehicle, trying to wrap their head around the concept of $/min * charging rate * wh/km is enough to make people frustrated and confused. Pricing per kWh is what people are used to when buying gas, and allows an easy calculation of what a full charge will cost. Especially since exactly none of the fast charging options for Teslas charge at a consistent rate, varying the charging speed based on battery voltage and temperature, making it completely impossible to calculate how much a charge will cost before plugging into a $/min charger without first breaking out charging graphs and doing calculus.
Yes, right now most DC fast chargers are so heavily subsidized and cheap to use that this isn't really a concern for most current EV drivers, however as charger demand increases and the more price-sensitive income brackets start moving into EVs this issue will become a much larger obstacle to widespread EV adoption.