You are getting the full Skyrim experience and how it is meant to be.
Mods aren't Skyrim. They cheapen the universe and water it down. Tacky add ons and gimmicks.
Nothing from the mod community is meant to be in the game and does not represent the game.
It's like saying you don't feel you are getting the full movie experience from a film because you haven't read all the fan fiction.
Mods aren't canon or a part of the Elder Scrolls universe.
Now that that is out of the way, mods can add a TON to a game and give it all the polish and extra features that will make Skyrim feel like a husk of itself without it.
That depends on what you define as "skyrim experience". Some people think playing the vanilla game is good enough, some bought this game mainly because of its capability for modding. I would not have bought this game if there's no Creation Kit.
To me the "skyrim experience" includes modding.
A lot console gamers could not understand that there is no right or wrong or compromise in modding. Take killing moves for example, some people like it, some people hate it. Some people want all last hits to show a kill move, some people want 50%, while others don't want any. Bethesda cannot make three versions of the game, they can only compromise and make one that suits the preference of most people or suits that of their designing direction.
However, when there's modding, people don't have to compromise. They can take whatever they want and just ignore what they don't like. Everyone can customize the game to their own likings and no one has to sacrifice or compromise (almost) anything due to what other people think.
As for the universe, we all know the universe in Elder Scrolls games is fake. It's just one story made up by Bethesda. Again, there's no right or wrong or what should or should not be here. There's only "what you can think".
Without mods, what you can think is decided by what Bethesda wants you to think. With mods there's almost no limitations.
Just remember, the universe in Elder Scrolls is not real. Without mods there's only Bethesda's Elder Scrolls universe. With mods there could be Tom's Elder Scrolls universe and Peter's Elder Scrolls universe. There's no saying that your universe is wrong and mine is right or that is cheap because it's not canon.
To me, in single player games the most important thing here is that it's canon to yourself.