What the built-in recorder does well
Press ⇧⌘5 and the Screenshot toolbar lets you record the whole screen, a window or a part of it, with a microphone, a countdown and a circle around your clicks (Apple's guide). QuickTime Player can do the same and trim or split the result. It's free, it's fast, and nothing gets uploaded.
What it doesn't do
The recording is exactly what was on your screen, at full size. On a phone, a 13-inch Mac screen shrinks to a third of its width and the button you're talking about is a few pixels tall. There's no zoom, no cursor clean-up, no camera bubble in the video, no captions and no link to send.
What Dolly adds
Dolly records just as easily (⇧⌘2), then does the camera work: it zooms in on your clicks and typing, redraws your cursor so it glides, and places the recording on a background with padding and a shadow. Add your camera as a bubble, cut silences in one click, make captions on your Mac, and export an MP4 or a GIF. With Get link, a short link is on your clipboard in under a second. While you record, you can read from a teleprompter or draw on screen.
It's free with no time limit; free exports go up to 1080p with a small "Made with Dolly" logo in the corner. Pro (€59 a year) removes it.