Some more recently discovered (for me) YouTube closed caption options.
If you click the three dot menu under the currently playing video...
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...and that video has closed captions, you should see a 'Show transcript' option in that menu, click that and all the closed captions should show in a transcript window beside or below the video.
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The captions will highlight in sequence as the video plays and if you click any one of the captions, the video will jump to that position. There is a three dot menu at the top right of the transcript window where you can toggle the timestamps on and off, I guess that makes it better for copy pasting the captions should you want to.