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Of course, they will still make a genetic contribution to produce decent egg-layers. Your plan is practical.
When I have asked this question, usually I hear Buff Orpington or Brahma, but the more docile roosters won't protect from predators with the same gusto as a more middle of the road rooster. Usually, Orpingtons lay more than Brahmas. But both are good for cold weather.
I have had good rooster experiences with green eggs layers (hatcheries call them Ameraucana/Araucana, have pea combs) and Black Copper Marans. Green egg layers are usually very good layers, while the Black Copper Marans no so much, but a cross will do better. Just my experience over the past few years.
I have had more trouble with Barred Rock roosters than anything else. I have heard so many horror stories about Rhode Island Red roosters that I've been afraid to even try them, same with Leghorns.
Truly I am surprised by our Black Copper Marans roosters. They're almost like puppy dogs, except I don't pet them or hold them or pick up hens in front of them, etc. I don't handle them except at night if I need to trim spurs or nails or look them over.
Truthfully, "ease of handling" isn't something I think of when I think of any breed of rooster. Once jumped by a rooster, always gun shy.
As for cold hardiness, look for anything with a pea or rose comb. Say a Buckeye or a Wyandotte. That way you won't have to smear his comb lightly with olive oil when it gets down below 20 in the winter (temp depends on the humidity and ventilation in your coop). The olive oil doesn't always work either. Both the Buckeye and the Wyandotte will do well in a cross with Production Reds.
One thing you should know, though, is that so far the roosters I've had have always been raised with the hens they've grown up with. Make sure he's at least 5 months old or he may get hen-pecked and then never be on any use. Another, I personally would shy away from any rooster that was raised to be a pet and carried around and petted, that then turned rambunctious.
Okay, that's my opinion. It's sad, though, that you will be able to find opinions that will contradict mine on every point. Each rooster (and how he was raised) is so different and my experience will not mirror someone else's. And views on crosses aren't always the same either. Hopefully, more people will chime in and offer their opinion. Eventually you may get a vague concensus.