Some facts are correct, some are not, and the conclusions are murky.
The association of Vitamin D as a cancer-preventative has numerous scientific literature backing it including
this paper that has additional references.
When he goes off into sunlight, and the associated Vitamin D, helping with all kinds of diseases and colds, then that goes way too far and is not supported by strong evidence in all of his examples.
I think a much more valid summary would be that quite a few people have Vitamin D deficiency and that they need to either take a Vitamin D3 supplement or spend 15-30 minutes in the sun a few days a week in the summer without sunscreen (or in the winter in the south, but in the north more exposure time is needed during the winter). He is correct that those with darker skin need more exposure if they don't take supplements (unless they live in intense sun areas).
He is incorrect to assert that if you spend more than this relatively small amount of time in the sun that you don't need sunscreen or clothing or otherwise avoiding such sun or tanning booth exposure. Your body will only produce as much Vitamin D as it needs and any additional sunlight just increases the risk of damage from the UV. UV damages organics and produces free radicals. Anti-oxidants combine with such free radicals to prevent damage, but this is a race and having more UV increases the statistical odds of getting cancer, aged skin, etc.
He is also incorrect in implying that you only risk skin cancer if you burn, but not if you tan.
So while it is true that having better nutrition is healthier, it is not true that one can spend as much time in the sun or in tanning salons as they like so long as they have adequate Vitamin B and related nutrition. What he said in the beginning about there being multiple factors is correct, but just because you improve your odds of better health with better nutrition doesn't mean that you don't decrease such odds with excessive UV exposure. Even studies of Vitamin D deficiency show a 50% reduction in melanoma when Vitamin D and calcium are at proper levels, not a 100% reduction.
Remember that evolution influenced skin color (melanin amount) over time based on where people lived so that those in very intense sunny areas through most of the year (such as near the Equator in Africa) had darker skin that produced Vitamin D more slowly and tended to block UV more to prevent damage. Those in northern climates had pale skin to absorb more UV to produce more Vitamin D since sunlight was more scarce. In our modern world where people move around to different areas, then they can get too little or too much UV absorbed with the associated problems of too little Vitamin D or too much UV damage. Proper nutrition helps to minimize the damage, but does not eliminate it.
As for why dermatologists and other doctors are so focused on skin cancer, it is mostly because non-melanoma skin cancer is the number one cancer in the U.S. which accounts for about half of all cancer diagnoses or about 2 million per year. Not all untreated non-melanoma skin cancer is fatal (about 0.1% for basal cell carcinoma and <1% for squamous cell carcinoma), but some is and left untreated can lead to significant local destruction and disfigurement and there is no way of currently knowing which will be so all are treated (removed). Lifetime cancer risk for the much more serious melanoma is around 2% while dying from it is less than 0.5%.
From a practical point of view, the main effect from UV exposure is aging skin since the tanning effect can be uneven and long-term exposure can weaken connective tissues. The main driver for sunscreen protection is probably more for maintaining "youthful" skin than it is for preventing dying from skin cancer. Given how relatively long such skin problems take to develop, it is most important to not overdue UV exposure when one is younger. This is similar to not smoking when one is younger. If one wants to do such habits with far less of an effect on one's health, then do them when you are much older. I think a lot of young people following Mike Adams' advice (i.e. go ahead and use tanning beds) are going to regret how their skin looks when they get older, even if they have proper nutrition which is often not the case. I frequently refused to wear sunscreen as a child, sometimes burned/peeled but mostly tanned, and now have skin that my dermatologist calls "scary".
). He never sunbathed in his entire life, but he did spend a lot of time walking up mountains as a young man, presumably with his shirt off. I suspect that genetics trumps nutrition any day.