I'm not sure what you're saying. I think you're saying christmas is a shortest day celebration, and that therefore in the southern hemisphere christmas should be observed in June. I think you're saying the southern hemisphere is doing the right thing on the wrong date.
If you never said 'christmas == cold' (two equal signs, note), you certainly did say 'christmas == shortest day', and by virtue of physics 'shortest day == cold'.
Seasons are local effects, but many inwardly-focussed societies don't grasp that. Whenever I see an ad that says "coming this summer", my immediate reaction is "when the fuck is that?". I'm not being obtuse for the sake of it, I'm just being tripped up by someone using a metric that is not appropriate for indicating time. I have to consider a) where I am, b) where the ad writer was, c) whether the ad writer has a global view (usually not), d) convert.
What do you in the north do to celebrate the shortest day of the year in the southern hemisphere? I'm guessing not much (and rightly so) !