Is there a color palette in LyX (similar to that in MS Word)? I frequently use colors and the quickest way I know of doing it now is to select text, click on the Text Style button, select color, and click "Apply" or "OK".
You can create a shortcut for a specific color, as answered here. Or, if you want to just apply the last text style, you can click on the toolbar button that has the tooltip "Apply last".
Other than those specific case (e.g., if you want to use many different colors often), it does require the steps that you already mentioned in your post. The reason for why LyX does not make it easy to do with two clicks (like, e.g., Microsoft Office and Libre Office), I believe, is that for most users of LaTeX it is more common to set colors globally, and to avoid "finger painting" text manually. Thus, the LaTeX (and LyX) way of using color would be to set a style, e.g. "important", where the color of "important" is, e.g., "blue". This way, if you (or your editor or co-author) decide later on that "important" text should be "red", you can easily change it. For more information on this philosophy (i.e., the principles behind the design decision of LyX), you can see the Wikipedia page Separation of content and presentation.
Note that in LyX, it is easy to set the custom style of text with two clicks (e.g. in the layout box, or Insert > Custom Insets).
All that said, I think it would be a valid feature request to LyX to have an easier way to set the color. The above explanation is just meant to explain why this (to my knowledge) has never been requested before.