Out of curiosity, what was the fault in that faulty heat lamp. That might help someone else avoid a fire.
What ambient temperatures are you looking at when you brood? The extremes are what are important not some average. The middle of a Michigan winter is quite different than the middle of your summer. It helps to know what you are working with. And how many chicks?
An outdoor brooder needs the same things as an indoor brooder: food, water, protection from predators, and protection from weather. Weather means rain, wind, and you may need heat. Weather is what you are asking about.
To me the best brooder is one that has a spot warm enough in the coldest temperatures and a spot cool enough in the warmest temperatures. In a climate controlled house that can be pretty simple but outside where the temperature can swing 40 or 50 degrees F in a day it can be more challenging.
You can use a heating device that heats the entire area. I understand heat lamps are out for you but heat emitters or radiators can be used. You can keep an area cool enough by having a brooder big enough and with ventilation so the far end can cool off. This is what I do, some mornings there might be ice or frost in the far end but the area next to the heat is warm enough.
Or you can use something like a heat plate or heating pad to literally create a spot that is warm enough. As Huntmaster said the heat plates have limitations in cold weather but in warmer weather they can work great. I'll include a link to Blooie's thread about the heating pad that Azygous and Rosemary use. Those have been used in very cold weather. Another option is the hover, if you do a search on "Ohio brooder" it should come up. Think of a shallow box raised so the chicks can get underneath with the open side down. Warm air rises and is trapped under the box.
The reason I asked about number of chicks is that some things like heating pad or heat plate have limits while the others may be able to handle more chicks.
Mama Heating Pad in the Brooder (Picture Heavy) - UPDATE | BackYard Chickens - Learn How to Raise Chickens
Good luck!