Think on it. The need for highly skilled surgeons would be less in a society where Restoration magic can heal wounds and purge diseases. The idea of a Fortune 500 pharmaceutical firm is laughable in a society where various plants can be used for everyday household needs.
Why develop refrigeration when magic can provide Frost effects? Why produce matches when everybody knows how to cast Flare?
Who needs light bulbs when you've got Light or Night-Eye effects?
The problem that extreme class-centrism presents is that Tamriel is not our world, so the adaptions that exist in the mechanics do not fully register in the mind.
As for the idea of not being able to perform certain magic if you're a pure fighter: to what extent?
In a world without something as basic as the Band-Aid brand bandage and Neosporin, what idiot wouldn't walk around with the ability to at least heal minor wounds?
In a world without the Zippo lighter, what moron wouldn't know how to cast Flare?
Sure, those people could exist in Tamriel. But to find them, all you need to do is visit the local graveyard, and find the section set aside for "those idiots that didn't realize even some basic proficiency in restoration is a good idea."
If you want to make your game harder for yourself, feel free to avoid any use of a skill if it's "not in your class." I won't force you to use it.
But don't think that it improves immersiveness for me. My characters have traits like common sense, basic reasoning skills, an understanding of logic, a willingness to use that brain they were given, etc. Bruce Lee was a great Martial Artist because Bruce Lee was a genius Martial Artist. And he also felt fine, nay, encouraged, using a gun to defend yourself if you could. Why willingly handicap yourself? In Skyrim, you're going to have Dragons trying to EAT YOU!
As for a requirement that you receive training to use a skill...well, how so?
I was two grade levels ahead in high school math and science because I taught myself algebra and biology straight out of a textbook. If only there were skill books in The Elder Scrolls...oh, wait, there are.
I went through school and paid tuition so I could take classes to prepare me for anything that might come up in life. If only you could find knowledgeable people in Tamriel and pay them for training...oh, wait, you can.
I took martial arts lessons and practiced for hours every week. If only in The Elder Scrolls you could spend time practicing what you have learned to make yourself better at things...oh, wait, you do.
As far as actually gaining proficiency in multiple fields, ever hear of the term polymath or Renaissance Man? Imagine, for a moment, that some people enjoy playing such characters in video games. ElderScrolls lets you do that. Why take it away, when it's really only an option.