The gallery needs polarising filters on their lights. That would cut down on the reflections. Alternatively, the paintings could have been framed behind polarised glass. If you look at the work through polarised sun glasses, you won't see the reflections. Light is magic. :)

The moment a smartphone manufacturer adds an automatic polarising filter into (its optical, not digital) into a camera lens, they'll prevent reflections like these ruining images. :)

Sorry for my fixation on the reflections as opposed to the work and your experience of it!