We need to be realistic here. We know what modern entertainment looks like and its not realistic at all to just 'read books' and play board games.
I don't know how much energy Netflix uses serving a movie, but playing a video game on my PC for two hours where I'm located might generate a kg of CO2. That's about as much as I'll breathe in a day. Relative to other sources of atmospheric CO2 I'm not that concerned.
I agree with your second paragraph, and selling the "make better choices to save the world" argument is an industry playbook favorite. Environmental damage needs to be put on the shoulders of those who cause it, which is overwhelmingly industrial actors. AI is not useful enough to continue the slide into burning more fossil fuels than ever. If it spurs more green energy, good. If it's the old "well this is the way things are now", that's really not good enough.
Driving too the cinema to watch a movie produces more CO2 than watch one movie online but online makes it more convenient so you watch more. That sums up to more CO2 emission.
The point is that higher efficency is wortless in terms of CO2 emissions if it leads to higher usage that compensates for the savings.
If a programmer can program faster with AI it's good if he only needs 1 hour instead of 8 but if he still programs 8 hours a day AI's energy consumption comes just on top of his previos consumption.
Climate change doesn't care how efficient you produce more CO2, more is simply more.
For everything else, there are already plenty of energy saving mechanism build into the CPUs, Mainboards, Disks etc. A Datacenter doesn't run on 100% Energy just because the load is reduced.