None. They'll certainly cite it in support of their position, but it is not the basis for it in any way ... since this could, even if authentic, refer to some future period of the Church.
Please read my other thread about this topic.
Your quote is what I believed yesterday about La Salette and its prophecy.
Today, I learned that AT LEAST ONE VERSION OF the La Salette prophecy, or at least chunks of it, was actually proven FALSE in an irrevocable manner (as in, unless God rewinds time or contradicts Himself, this prophecy can't be fulfilled).
And another version of it, with no egregious errors I could find, shows a prophecy that IN NO WAY could refer to Conciliar Rome and Pope Francis today.
So either way, a lot of Trads have the wrong idea about the La Salette prophecy!