The kernel trick uses an implicit mapping into a higher-dimensional feature-space. On the other hand, Factorization Machines uses an explicit mapping into the polynomial kernel space. However it learns jointly the "right" polynom, by mapping the base features into a low-dimensional dense space where the higher-order terms of the polynom are dot-products (we're not taking dot-products in the original feature-space!). FM then learns the right polynomial (a non-convex taks) jointly with learning the original supervised learning task.
Where I am not with you is in the hint that kernels have feature maps that are necessarily opaque. In fact inhomogeneous poly kernels ar great examples where the feature map is known in closed form. Although such a map is available it is not always efficient to use it. But what it lets you do is optimize the weights of each dimension on tje mapped feature space but executed in the native space. In fact, and I am greatly tickled by the coincidence, I had sent a mail to colleagues where I suggested a form where instead of the standard Euclidean dot product in the poly kernel is replaced by a convex combination of other favourite kernel of choice. The training algo remains almost the same. I was not aware of this piece of work but the mechanism is pretty much the same (little more general. Essentially replaces the std dot product that appear in the kernel expression by other kernel expressions. If you are thinking recursion now well that's intended) and can be pushed further.