Absolutely not. Krylov methods are one method for solving a linear system and can absolutely be used with stochastic optimization methods. Generally speaking, most optimization algorithms require the solution of some linear system and we can use a Krylov method or we can use a direct solve or we can integrate the direct solve with the Krylov method as a preconditioner. However, this has nothing to do with the stochastic piece of the algorithm. If we want to use a random projection or some kind of other technique to compile the gradient, we can absolutely do so while using a Krylov method.
Now, if we're just doing a stochastic steepest descent algorithm, there are not really any linear systems to solve, so this doesn't come up. If we want to use a stochastic Newton algorithm, this does. It comes up here specifically because forming the full Hessian may be too expensive in terms of computation or memory, so if we're smart and use a Krylov method that only requires Hessian-vector products, we can effectively use second-order information without taking the penalties of computation and memory. Yes, there is a loss here since we only use a piece of the spectrum of the Hessian and this is where the art and science and preconditioning come to play to insure that we use this effectively. However, we can use stochastic algorithms here. In fact, the entire field of stochastic PDE constrained optimization deals very precisely with these tools and does so in a surprisingly effective manner.
The advantage that Krylov methods have over a direct solve is that we retain the ability to use the direct solve if we want in the context of a preconditioner. However, different Krylov methods have different properties that we can use to our advantage. For example, conjugate gradient works on a minimization principle based on a quadratic form. This integrates well in trust-region methods that require us to minimize a quadratic model. GMRES minimizes the residual of the linear system, which gives us a monotonically decreasing residual at each iteration, which can be important for different algorithms. Which Krylov method to use in any particular situation depends on the properties desired and the systems involved. And, I can't stress enough, we don't lose out on direct solves. Direct solves can be fantastic preconditioners and if we really can compute an inverse exactly our Krylov iterations converges in one iteration. As such, we lose a little bit of computation and memory, but not that much.
Anyway, sorry for coming on a little bit hard, but I wanted to try and nip some amount of confusion in the bud quickly. Stochastic methods are naturally and fully integrated into inexact optimization schemes that rely on Krylov solves. They are the standard tool for many fields that use optimization. They may not be wildly used in ML, but that has little to do with their efficacy computationally.