You could have $A\cos(\omega t) \sin(kx)$ and talk about the ‘amplitude’ of the time dependent function $\cos$ if you wanted to. It’s a matter of interpretation, the result is the same.
To see this, take out the time dependence (scale it to 1, the same as considering the wave at time $t=0,2\pi,4\pi,...$). Now you just have a wave in space.
As you change $A$ bigger or smaller, you make the wave in space taller or shorter. If you set $A$ negative, then the wave ‘inverts’ and what was positive becomes negative and vice versa.
Now imagine that you move $A$ back and forth, growing and shrinking your wave and inverting it, with a regular motion that can be described in time by $\cos(wt)$. This function is $A\cos(\omega t)$. So then we have added time dependence back in.
What the function $\cos(\omega t)$ does is it scales each point in space by a factor (between -1 and 1) that depends on time in a certain way. So each point in space ‘waves’ up and down between its maximum value and its minimum value.
If you thought of $A$ attached to $\cos$ rather than $\sin$, then the time dependent function scales each point in space between $-A$ and $A$. So the scaling with time becomes stronger if $A$ is large, and weaker if $A$ is small.