Is this really an expensive problem to solve?

Give Google a continuous feed of the weather data which they cache locally. I can't imagine that being a particularly expensive thing to operate - no need to reply to an API call from Google every time someone searches for "weather".

As long as it’s properly addressed to avoid abuse, I don’t have a problem with private companies and individual citizens both benefiting. You could easily put rate limits if you think it’s a major issue while still maintaining the free service for smaller users. I personally don’t like the privatization of profits while also maintaining the narrative that companies don’t benefit from public works.

Sure, why not. No problem.

That's a good example of how government open data can support both people and business.