I think so. It's very rare that I need to paste with formatting. The majority of time that I actually use normal paste is when the source content is either of the same format or has no format already.
Word at least lets you choose the default action and offers a little contextual toolbar on paste to change the behavior for that pasted snippet as well. The options are iirc: Paste with original formatting, paste with formatting adjusted to document styles, and paste without formatting.