Ok, look, I can give my two cents (as a linguistics major).
I can pronounce nuclear both ways just fine (same for nucleus). When I speak faster I am more likely to say “nukyular” but I can say both at most speeds. (I’m from the Northwest).
The question has been answered as to why people “mispronounce” it. That answer is that the cluster of consonants in that context (i.e. “u-cl-ea”) is not simple. Some languages don’t ever have consonant clusters, there’s always a vowel between individual consonants (though the concept of ‘individual consonant’ is a little bit misleading).
Because consant clusters are hard, especially between two non-reduced vowels (i.e. those that are not schwa-ish), there are some dialects in English that find it very difficult to say the word nuclear and resort to the common tactic of inserting a reduced vowel (i.e. a schwa). This is entirely natural and common. You most definitely do it. It really doesn’t matter what your dialect is, chances are very high that there is a word for which you insert a vowel that isn’t in the standardized spelling. Listen when you speak and you’ll find it.
The theory that we shortened the word to ‘nuke’ and then worked from there is almost certainly not the origin of this pronunciation but may have some effect on it. For example to reinforce it. This is a sort of Occam’s razor thing, it is just a simple answer to hypothesize an ease of pronunciation issue.
As to the issue of written vs. spoken, I have a few thoughts on the matter. First of all, spoken language came first and even today serves as the medium for the vast majority of human communication. Just think of all the conversations you had today, just think of all the people who are illiterate or semi-literate. In that sense spoken language is vastly more important.
Written language doesn’t have to have a particularly close connection to the pronunciation of spoken language. Look at Chinese which doesn’t really encode sound (at least not for meaning). Look at Hebrew, which doesn’t encode vowels. Look at English spelling which isn’t too consistent in its spelling but still basically works.
In some sense the idea that spelling doesn’t have to reflect pronunciation is an ideological stance, but it is one that is backed up by some reasonable evidence (for example, what I’ve said above). The fact that we can still read Shakespeare without too much trouble attest to usefulness of keeping your spelling somewhat static. Furthermore, the regional variations in English are wide enough that it is probably easier to just spell everything in the standard way rather than try to reconcile the irreconcilable pronunciations in a spelling system.
Having said all that, I must admit that in theory you ought to be able to make everyone pronounce things as they are spelled in Stamdard American English today, but it would require a lot of work and probably engender a certain amount of resentment. No one would be spared from having to change and some words would be pronounced in ways that no one pronounces them currently. If you wanted to keep everyone pronouncing things this way you’d have to expend a lot of energy to keep people in line phonetically. We’d all sound basically the same, at least in terms of accent. I think that those people with speech impediments would have an extremely difficult time, much worse than today.
As for our president’s use of the “nukyular” pronunciation, you can certainly be amused by it, but it is not bizzare or abberent nor is it in and of itself really an indication of low intelligence or bad breeding (so to speak). In our society we do value a Standard American English dialect, which I believe includes “nuclear” as a proper pronunciation of the word. So in fact it may say something about our president’s intelligence and education that he has not learned to pronounce nuclear in this way, in that he may be incapable of pronouncing it or never learned the proper way. However, it is probably safe to say that he is aware of the standard pronunciation and whether he chooses not to or is unable to pronounce it in this way says nothing fundamental about his intelligence.
Oh, and Kennedy didn’t say Kuber if I remember correctly, but Kyu-ber.
Looking at this on Preview I see that it is a pretty big reply and I hope you’ll wad through it. I also hope it is clear.