@ZOS_Kevin not sure if this is intended, or if its even something that can be worked around (I am not sure how the weather mechanics work in ESO as far as an effect), but as the title states the Alinor Greenhouse Summer does'nt block rain.
The image below is of the Telvanni Alchemical station with some rain/water on it (hard to see in screenshot.)

This image just shows that the crafting station is inside the Greenhouse with the weather control icon at the other end on a table (glowing purple/storm setting)
"Class identity isn’t just about power or efficiency. It’s about symbolic clarity, mechanical cohesion, and a shared visual and tactical language between players." - sans-culottes