How is it nonsensical? If one compares equal time spent on each (which would be the only fair comparison), then generative AI consumes magnitudes more energy than a static web page like Mumsnet.
In fact, the need for generative AI data centres is a massive shift and increase in the impact of server farms – AI requires much more computational power than general internet usage.
"While traditional data centers are built primarily to support CPU-centric workloads with standard requirements for security, reliability, and energy efficiency, generative AI demands a fundamentally different infrastructure approach. AI data centers require high-density GPU clusters and advanced networking, storage, power, and cooling capabilities to handle the extraordinary computational intensity of modern AI workloads."
Also, as this article points out, "Unlike when you stream video or load a large web page, with generative AI most energy is used at their end, while processing your query."
Obviously, video streaming uses energy too, but not with the same intensified data centres that have much more (an ever increasing) impact on the environment and consumers. Besides, you were talking about Mumsnet usage – that's a drop in the bucket compared to the impact of using generative AI on a daily basis.
This Wired article is interesting too, regarding the impacts of data centres especially.