You have the basic idea. The missing piece is that in the end, the total tax calculation using the gains added to other income is compared to the tax calculation using the graduated capital gains rates and the lesser tax amount is used on your return.
The worksheet used to calculate your taxes when taking capital gains into account is found in the IRS Instructions for Form 1040. It is called the Qualified Dividends and Capital Gains Tax Worksheet.
You can find it on page 33 of the PDF document at this link: 2019 Form 1040 Instructions
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