Semi-transparent mirror works by both reflecting light on the same side and passing through light from the other side. For the effect to appear, one of the sides needs to be lit significantly brighter than the other. Thus on the darker side, light passing through will dominate, while on the brighter side, reflecting light will dominate.

So when protecting museum items with such mirror, they should be put on the brightly lit side of that mirror for them to still be visible. A flash would still make the other side brighter and thus the photo would capture only reflections.

The reason flash photography is discouraged is that light is a carrier of energy, and that energy affects the lit surface destructively. In this context, the aim is to keep displayed items in best possible shape for decades, or even centuries.

So such mirrors are not used in museums and galleries, because using them would require constant bright lighting, and that would be a counter-productive strategy for avoiding short light bursts.

Current strategies are the prohibition messages already mentioned in question, accompanied with generally dim lighting and obstruction of direct sunlight reaching the displayed items.

In aquariums and zoos, the concern is different - flashing light in nature usually denotes approaching adverse weather conditions and as such would cause unnecessary stress for the inhabitants. Thus no need to use dim lighting, or protect from direct sunlight (exceptions may occur depending on habits of the specific species).