My layperson's understanding regarding the criticisms against Julian Baumgartner is that he uses a lot of invasive methods that don't meet the strict technical standards employed by professional conservators, and this creates a misleading impression of what conservation actually is.

For example, you will frequently see Baumgartner do over- and in-painting of fairly large areas that have been lost. Modern conservation has slowly evolved to distance itself from mere restoration; the objective of conservators who work for museums or major collections is to only apply non-invasive procedures that can bring the artwork closer to its original state (e.g. grime removal) and shore up the structural integrity of the physical object to preserve it, but without adding anything non-original if possible. Baumgartner claims that all his changes are reversible, but you still see him prying lacquer off with a scalpel, completely replacing the wood of wood-panel paintings, and many other techniques that cast doubt on those claims.

I'm not a professional, so I can't judge whether all of this is accurate, but I find the drama fascinating. In the field of conservation, Baumgartner is an outsider, as he mentored under his father rather than study conservation in an academic setting. Baumgartner's videos look absolutely world-class, but he probably wouldn't meet the bar for qualifications imposed by major museums, which typically require a postgraduate degree in conservation, including formal training in chemistry. So there's probably an element of disdain for amateurs to the criticism, as well.

To be clear, what Baumgartner does is probably perfectly acceptable for many artworks, where the owner's directive is to restore the artwork back to what it may have looked like when it was made. But I'm not sure he should touch a Vermeer.