The BBC is publicly funded. The entire idea behind publicly funded news and broadcast is that they shouldn't care about eyeballs, singe they don't need and money.

The BBC is staffed by humans. Humans who want their programming to be popular and watched widely. If people stopped watching BBC news, you better believe they'd make changes to reverse that, whatever it takes. Their incentives are just different.

If no one was watching or reading I can't imagine they wouldn't start facing political pushback as a waste of money. Every entities existence is strengthened by usefulness, profit seeking or otherwise.