Then what are people referring to when they say "Capitalism caused this"? If capitalism is private ownership of tools, and there are systems other than capitalism with private ownership of tools, what are people diagnosing when they say 'capitalism' in this context?

Well capitalism isn’t just private ownership of tools, it’s also division of labor, wage labor, and market exchange. Possibly some other things that are escaping me at the moment.

The critique ultimately traces back to Marx, who asserted that social relations within a society are dictated by the “mode of production”.

I don’t necessarily alway agree with people when they say “capitalism caused this” but this is by best attempt at relaying the reasoning.

Ultimately the critique that socialists are making, to bring it back to your analogy, is that most people don’t own their own hammer, capitalists do, they just do all the work with it to build houses while the capitalists take the lions share of the profits.