First of all, thanks for keeping the discussion on the changeset civil despite a hot-button topic. There were plenty enough things for OSM to debate before all this started. 
The mountain did come up in the Gulf thread, but it got buried pretty quickly. Most of the community is international, so they care a lot more about the Gulf.
There are some general practices around name tags that we should maintain regardless of where we land in this particular case:
name=*should match thename:*=*in the predominant local language. That would bename:en=*, unless we really want to list out the name of the mountain in each of Alaska’s 24 official languages, separated by semicolons. (Which we could do…) The purpose ofname=*is for data consumers that don’t or can’t personalize the map to each user’s preferred language on the fly.name=*andname:en=*should be the predominant name for the feature, or at least a very common one if there isn’t a predominant name. Any other fairly common name can go inalt_name=*and other*_name=*where geocoders will find it.
Normally, official_name=* is where a name would go if the authority in charge of the place has designated a name in spite of common usage. This feature had an official_name=*, but I removed it in favor of nat_name=*, reg_name=*, and loc_name=* after there seemed to be consensus around this proposal. Since then, Alaska has continued to assert Denali as the mountain’s official name, as it has for decades, so we essentially have a geopolitical dispute between a country and one of its states, not nearly the first time this has happened.
Since the mountain is in Alaska, I think the national/regional/local scheme is adequate for clarifying the scopes of the two names. This doesn’t mean the names aren’t official. However, this approach could prove inadequate if other states or countries begin to designate their own names for the mountain in English. So far, the only other state I’m aware of that has taken a stance is Oklahoma, though it’s unclear if the official who made that pronouncement ever put it in writing. Iowa and Tennessee also have pending legislation about it. If these efforts succeed, we can switch to the same solution as with the Gulf: official_name:en-US=*, official_name:en-u-sd-usak=*, official_name:en-u-sd-usia=*, etc. All of these locale codes are based on fairly obscure provisions in the technical standards we’re following, so some data consumers will need to fix bugs in their parsers to recognize these tags.
As of version 67, name=* and name:en=* are both set to Denali, while Mount McKinley appears only in nat_name=* and old_name:en:1896-2015=*. As both of these dates are somewhat misleading or at least lack the necessary nuance, I’d favor moving Mount McKinley and its translations from old_name:*=* to alt_name:*=*, where they’re more likely to be accessible to geocoders. I’ll let others weigh in on whether we should then swap name=* and alt_name=*, but I’d personally place some weight on the local situation, since OSM has a fairly unique focus on mapping what’s on the ground, even despite other maps usually say.