A Boring Dystopia

My question is, wouldn't the power needed to run these negate the benefits they bring?

The hardest part about green energy is getting it to the time and place where it can be most useful. That's why real time solar power prices sometimes dip negative (where the producers are literally paying people to take that excess power off the grid), and sometimes in consistent and predictable ways (e.g., California's "duck curve" in spring and autumn).

So with solar power being the cheapest form of generation, but highly dependent on weather conditions, the solution might be to build up overcapacity where production during cloudy days is enough, and then find some way to store the excess on sunny days for nighttime, and maybe using intermittent power sinks that can productively use energy only when the production is high (charging batteries, preemptively cooling or heating buildings and storing that for later, capturing carbon, performing less time-sensitive computer calculations like data analysis for science, etc.)

If we have systems that produce too much energy, then carbon capture (including through manufacture of fuel or other chemical feedstocks) can vary by time of day to address overcapacity.