Here in Florida we have similar problems with Brassica. These are cool weather plants and actually enjoy a touch of frost, once they have a few leaves. I think your problem is the temperature more than anything else. I grew Brassica - broccoli, cauliflower and cabbage, successfully this winter for the first time since moving to Florida. Of course, I think it helped that we had one of the coldest winters on record. It actually SNOWED one day! Well, sorta.
I generally start my seeds by simply broadcasting them in a shallow tray, about 2 inches (5 cm?) deep, filled with good potting soil. I make sure they have good drainage and never let them stand in water. Once they have their second set of leaves, I prick them out with a tiny trowel and pot them up individually in 3 inch (7-8 cm) pots. At this point, they can handle a light frost.
I'd let them get a bit of a chill, if possible, but don't let them freeze.
A better way to start your Brassica next year may be in a cold frame. Do you use such things in your part of the world?