I'm going to definitely vote outdoor brooding. Young growing chickens produce a TON of dust. One of many newbie mistakes I made when I first got chicks in 2009 was brooding them in the basement. The chicks did great, but the dust they're capable of producing can threaten your sanity, LOL.
I just got some hatchery chicks Friday, and I have them set up in a stock tank up against the house, under the deck. The overnight lows have been in the low 50s. I have the front wrapped in a heavy blanket for insulation, and another blanket covering the top during the night, and part of the top during the day. With the heat lamp in there, it's nice and toasty for them, and there's no dust or noise in the house. And there is also a homemade custom-fit hardware cloth lid for the tank as well, to keep predators, particularly the barn cats, out.
The main concerns with brooding them outside is keeping them warm enough, and keeping them safe from predators. Pros of course is no dust or noise or smell in the house. I also like that my brooder happens to be close to the garden hose, so I'm not tempted to use a sink for refilling their water. I also like that when they're about 5 weeks old I can easily start getting them accustomed to the overnight lows (assuming the temps are reasonable of course) and then can move them into the chicken tractor, so they can learn to eat greens and bugs from a young age. By keeping them on the cool end of their comfort range, they feather in faster, and so can be moved onto pasture in the chicken tractor sooner. The sooner they can get out on pasture, the better foragers they'll be as adults. Indoor-brooded chicks are more pampered temperature-wise in comparison.
Pros of brooding indoors is that you can get chicks year round. I would hesitate for example to brood November-February hatched chicks outdoors, just because it could be difficult to keep them warm enough. Cons of brooding indoors include the dust (this is a biggie, the amount of thick heavy dust they can produce is insane, at a minimum you'll want to cover an indoor brooder with a sheet) and the noise and smell.
Your climate is very similar to mine, so late May chicks should be quite easy to keep nice and warm, even in an outdoor brooder, so long as it is free of drafts. Just keep them warm, safe from predators, and like you say, confine them at first so they're never too far from heat, food, and water, and they should do great for you.