You assume that you can separate the data linearly in a lower dimensional space after some nonlinear transformation.
Kernel methods are popular due to the exact opposite: the data might only be linearly separable in a higher dimensional feature space. For any data you provide a higher dimensional feature space exists in which the data is linearly separable. Ofcourse, you don't need to go to a higher dimensional space for all problems but it is often necessary.
What you are interested in is VC dimension. The VC dimension of a model $\mathcal{F}$ is the maximum number of points that can be shattered by $\mathcal{F}$. The VC dimension of oriented hyperplanes in $\mathbb{R}^n$ is $n+1$. This shows that, for any amount of data points, a linear separation is always possible when we go to a feature space of sufficient dimensionality (but not lower!).
Kernel methods provide an efficient means to embed data in such higher dimensional feature spaces at low computational cost thanks to the kernel trick.
Explicitly transforming the data to a space in which it is linearly separable is not always possible. This is easy to see when you recall that some kernel functions, such as the popular RBF kernel, compute inner products in an infinite dimensional feature space. Obviously, you cannot create an infinite dimensional vector yourself.