Discrete State Space Model - Why Are We Calculating $ x \left[ k + 1 \right] $ Instead of $ \dot{\boldsymbol{x}} \left( t \right) $?
I will ask you something that will give you intuition.
How would you calculate the Gradient of an image?
Image is a discretization of reality, so how would you estimate the gradient of the "Reality" if you're given only the image?  

In the case above we use Finite Differences to approximate the continuous derivative.
So what actually is approximating $ \dot{x} \left( t \right) $ is:

$$ \dot{x} \left( k \Delta t\right) \approx \frac{ x \left[ k + 1 \right] - x \left[ k \right]}{ \Delta t} $$

I do a little abuse of the $ k $ parameter yet I think you can understand what is going on there.