We would have to file to get the proper credit, though, right?

The W4 form should be sufficient (since it's used to determine the credit, if any, you receive per paycheck). You shouldn't need to file a 1040. Your employer would file the W-2 form which would allow the government to determine your income for the purposes of the amount of credit you receive per paycheck.

So any income other than as a W-2 employee doesn't count influence your credit?

This includes capital gains, rent collected from tenants, AdSense revenue.

The right way to handle this stuff is not immediately obvious to me.

Isn't that income also reported to the government via 1099 forms or similar? Do those who make less money typically also have income from sources such as collecting rent or capital gains?

I suppose that reporting does happen or at least could be made to happen under the new system.

I used to make $80k per year in salary(which is far less than most of the top decile in the USA) and in one of those years had over $10k in capital gains. In the same year I thought about buying a house to rent out. I ultimately didn't do it, but if I had, then I would have had to declare the income as part of my tax return filing process. Just as one data point.