In my experience, a kid riding in a stroller wearing a rain coat gets SOAKED. It seems like riding in a trailer would be pretty similar. I vote for the cover.
I've never biked in the rain on purpose (got caught out once or twice). Is it really safe?
Absolutely! I'd argue that bikes are even safer, because you have vastly more visibility. In fact,I cycling safety sites usually caution being careful and being brightly lit more to compensate for the poor visibility of people in cars, not any inherent danger in your activity.
Bike tires hit the road as a curved surface. The tread on road tires is just decorative, it actually serves no functional purpose, because hydro planing is impossible. The only tricky spots for bikes are metal expansion joints and certain railroad crossings - you can lose traction on wet metal very easily. But with that small exception, you can bike more or less just as fast. Standard brake pads will lose a little grip, so if you start biking in the rain a lot, it's worth getting pads like the Kool Stop Salmon.
I guess if we NEED to be somewhere, I'll throw a cover on, but long term I want them all to get used to being outdoors. I think riding their own bikes in rain has a much lower misery factor than sitting in a trailer in the rain. Generally when I ride in the rain, I keep my torso dry but unless it's cold rain, I won't bother with rain pants or boots, instead wearing the fastest drying shorts I have and sandals instead of shoes.