Here in the Houston area?
Winter. The average high is in the 60s and getting up to the 70s or 80s is not uncommon
Exactly!
If you have a sealed space suit.
Absent that level of isolation from the environment your extremities and face and head will be miserable even if somehow your core is comfortable.
When I walked 4 miles to my office 4 days a week, I definitely preferred summer. Temps above 30C are uncommon, although not unknown. I did not walk when the temp fell below -25C. Even at -20C I would wear a face mask and it would be dripping with icicles by the time I got to my office. I would also not walk if the ground was ice-covered.
Winter by a long shot. The woods are beautiful in the winter, snow shoeing is a blast, and when the trails are packed there are no rocks or roots to trip on. The right clothes take care of the cold. No mosquitoes or ticks or other annoying critters.
This isn’t really correct. With appropriate clothing, and assuming you don’t have circulation issues, at a brisk walk your extremities will stay warm even in extreme cold conditions. The key is to generate enough body heat that your core is trying to dump heat by increasing circulation to those extremities.
It is a bit of a tricky situation - if you’re working hard enough to keep the fingers and toes warm, you’re probably also sweating a bit, so if you ever stop moving you’re going to be in trouble. I’ve been out walking for well over an hour in -35C and my fingers were only chilly the first few minutes.
I’m not going to criticize anyone for staying inside when it’s that cold, but it’s not difficult to manage the temperature assuming appropriate clothing (lots of wicking, breathable layers topped by something goretex, preferably).
I don’t have diagnosed circulation issues. But in cool to cold conditions my fingers and toes and ears & nose are always miserable no matter what gloves, socks, boots, etc., I might be wearing or whether I’m standing, walking, running, or whatever.
I lived in the US Midwest snow zone for 20 years. Never found a combination that worked.
I’m happy you’re built of stouter stuff. All my blood retreats the center of my body and that’s the end of comfort.
I live in South Florida. Absolutely want winter time walking.
I’ve walked many times in –30F. I leave the dogs at home when it’s that cold. Walking is my daily exercise, and I do it regardless of the temperature or the rain, except when the wind is an issue. Wind at 10mph makes a wind chill temperature of -53°F. Then stay home!
Perhaps a lifetime in Saskatchewan confers beneficial adaptations to the cold. Lamarckian inheritance is still good science, right?
My 2/3rds lifetime spent in the tropics or the desert has sure done that to me. Whether it’s heritable to my putative children is a separate question. 
I’ll suggest that part of it is preferential migration. The climate each of us in born into is essentially luck of the draw. Natives of whatever climate will tend to migrate to a climate more conducive to their personal preferences / body type / etc. Leading to the cold-hardy sorts living more poleward and the more heat-o-philic sorts living nearer the equator. Net of national boundaries, jobs, spouses, and all the rest.
If the last ~3 centuries of North American history had been different and the USA and Canada were side by side instead of stacked one atop the other it’d be interesting to consider how their respective populations would be distributed.
Hand and foot warmers are great if you want to walk in the cold but it’s too much for your extremities. Companies with hiking supplies carry them.
Can’t give a preference for an entire season. For me, a nice day in winter is vastly better than a hot day in summer. But a cold windy day in winter can be brutal.