The issue is less their buildings, rather their management of natural areas. That includes part of the coast and significant areas of moorland. National Trust management of these areas usually consists of building large car parks, paths and amenities next to otherwise empty natural areas then subsequently restricting access to significant portions after finding the area can't cope with the increased traffic. All in all usually meaning those that knew about that area prior to the "regeneration" come back to something that has been severely damaged.
Their management of moorland and hillside is similar, but also includes backing of damaging farming techniques and a the insistence of keeping "as is" rather than letting regeneration/rewilding happen*. This is especially prevelant on upland areas that are essentially monocultural wastelands with nature taking a back seat to grouse, deer and sheep farming.
The NT is a company like any other, a company that needs growing membership and increased footfall to buy more land. People have a rose tinted view of them, even though they also do plenty of good with old properties and (*admittedly increasingly in the last couple of years) some old estates - the "R" word is becoming less of a rude word to them.
I'm sure you'll disagree and that's fine, a lot of it comes down to whether you see the future of the UK as a continuation of heavily managed/damaged areas that are seen as natural (British countryside) with a significant detriment to most British wildlife, or you see it as Blakes green and pleasant land and would rather it didn't change back to what it was before human intervention.
The issue is, I'm sure it's not trying to do this, but it's a trust that's inherently conservative by design. Add to that it's mandate to allow people access to otherwise inaccessible areas and you end up with an organization that will inherently damage the landscape and position itself as a protector of "what is", rather than what it was and can be. In the last couple of years they have been dragging themselves up on the rewilding front, but I'm not sure how they can ever sort themselves out on the access issues.
Going back to the OP the issue is the NT own and manage a significant amount of the UKs coast and upland areas, as well as the old country estates (it's the second largest landowner in the country). That's a significant amount of land that their wholesale hand waving has now banned drones on. Most of which you could quite safely fly a drone in.
