The Pure Paste app (App Store link) by Sindre Sorhus removes formatting automatically when you copy and paste.
The newest app version requires macOS 14 or later. However, you can get the last macOS 13 compatible version from the app's website.
Pure Paste lets you paste as plain text by default. It sits in the menu bar and clears formatting (fonts, colors, bold, links, tables, etc.) from the text you copy. However, it does not touch unrelated content like files, images, etc. It also ignores content copied from password managers.
The app also includes other features (only indirectly related to the OP's question) such as "optionally preserve links" or "remove tracking parameters". You'll find more on that in the app store description.
I've been using this app for weeks now, and it works well for me. I used to use Flycut and paste with a Cmd + Shift + V
key combo all the time, but with the Pure Paste app I don't have to do this anymore. And whenever I do want to have formatting, I just right click and paste via the context menu.