>My question is: if we have this behavior not only for templates, but for every variable, isn't that less type-safety?
It depends on how you code, I suppose. I use var very frequently in C# now after being averse to it at first. I've never run into an issue where var caused problems in my code (rarely my IDE won't be able to pick up on what type it should be, but it compiles fine).
If you're changing a variable between compatible types, like a parent class, or changing to a different type of enumerable class where usage is exactly the same, you don't really run into issues. If you're changing the entire usage of a class (like from a vector to an array, or something even more different) than you probably should not simply re-declare the variable, you should be rewriting that code.
You wouldn't change
vector<int> numbers = new vector<int>;
to
int* numbers = new int[512];
Without making accommodating changes to the following code, would you? Then you shouldn't change
auto numbers = new vector<int>;
to
auto numbers = new int[512];
Without making changes to the rest of the code either. It's exactly the same as before, except you get to type less and it clutters the screen less when you don't need to write types multiple times on the same line. Especially for very long types names.