Yes, both platforms have native screen grab shortcuts that let you select a rectangular area to capture to your clipboard, which you can then paste into a messaging app.
Windows uses ⊞ Win+Shift+S (S stands for Screenshot)
Mac uses ⌘ Cmd+Shift+4 (4 stands for…
That said, I like the idea of being able to capture a text version of the chart to your clipboard too. It’s true that it might break down with some charts, and I’ve seen many different interpretations of text formatting for chords. It also tends to rely on monospace fonts, which aren’t necessarily available in a messaging app. But I have it on my list and I’m down to give it a try at some point when I get a chance.