I suppose it's because of its role in preserving and protecting national heritage. Its work, and the histories of many of the properties it owns, inevitably places it at the heart of current history wars. The relatively recent decision to, in some cases, present more challenging, previously overlooked histories has probably also challenged identities that some people hold dear. I'm sure the membership skews heavily middle-class, white, and older (this isn't a dig, I was a member for years). Going to visit National Trust properties at the weekend is something "people like us" do. It was cozy. See a nice house, have a cream tea, buy some pot pourri.
But overwhelmingly, the outrage is driven not by the membership but by dark money and the reason there is surely pretty simple: as a membership organization (rather than a government body) it is inherently vulnerable to takeover/capture. That Restore Trust has singularly failed to do that so far is a testament to the membership.