Eh, that isn’t usefully true: If the host fails to obey the takedown notice, they lose safe harbor, which is tantamount to inviting a lawsuit on very unfavorable terms. The takedown notice may not, strictly, be a ‘legal action’; but it compels a takedown on pain of substantial legal exposure.
Similarly, the counter-notice may not be a specifically legal document but (in addition to the fact that counter-notices tend to get process a lot more slowly than notices, making notices a decent DoS attack, filing a counter-notice indicates a willingness to uphold the validity of your original post in court. If you can’t do that, you have trouble.) The process is intimately tied to legal processes, even if it usually doesn’t go that far, ending in capitulation instead.