But in this case the negative externalities do not come from the mining but from the energy generation. So that is the level where the externalities should be priced in.

In the current environment, you can't separate energy usage from energy generation in terms of externalities. Electrical power is something like a fungible asset, so even if you are using 100% green power (which doesn't really exist), you're still increasing demand for electricity overall, which will result in more carbon-intensive power generation in general.

In other words, decreasing power consumption is as important if not more important than increasing clean energy as a share of production.

Electrical power is only fungible in that sense if significant negative externalities exist which allow units of energy to be priced similarly when their sources vary significantly in total (including societal) costs. If societal costs are internalized (i.e. externalities are eliminated) then fungibility would surely disappear, because a unit of energy produced from coal would surely be more expensive than a unit of energy produced from solar.

Is this a distinction without a difference? If the climate impact of mining were truly priced in, is there any chance this activity would be remotely profitable?

Of course it would be remotely profitable for some of the miners. The difficulty adjustment is making the consensus kind of invariant of the total amount of energy used. Even security-wise Bitcoin would probably be fine at 1/10 or even 1/100 of its current power usage