It sounds like a combination of temperature and nutrients. I think most of what you're seeing is probably typical for this time of year in our climate. AP and soil growing complement each other but I've found that the soil warms faster in the spring. I stick to cool weather plants in the AP well into June but even these I start outside of the AP system in trays and then transplant into AP. In the Fall the advantage seems to go to AP, the plants in my AP system hang on longer. I try to take advantage of each growing environment as much as possible. I do root crops and warm weather plants in soil. I keep the cool weather plants going longer into summer with the AP and extend the season. Once summer starts heating up I'll start putting some warm weather plants in the AP like melons, peppers, basil and bush beans.
I suspect that the water is too cool for the bush beans and for most warm weather crops still. For the broccoli, some varieties of broccoli only produce small heads. If you're seeing large heads in the soil garden and the packet you used wasn't a mix then I'm not sure what's going on. High temps can cause small heads on broccoli as can nutrient problems.
This could also be a light problem, any information on that? The surface of the bed may be getting too hot so just a bit of shade might help stop the plants from bolting.
If you have some pictures you could post we might be able to tell you what's going on, especially if there is a deficiency
Cheers
PS - What's your water temp running? Right around 60 seems to be where most things start growing a better.
edit 2 - looks like a temperature issue on the broccoli (either high or low) based on this document - https://extension.unh.edu/resources/files/Resource000605_Rep627.pdf