Intuition behind similarity solution for Heat equation

Question

The heat equation is $$\frac{\partial\theta}{\partial t}=\kappa\frac{\partial^2\theta}{\partial x^2}.$$ From this, one can look at dimensions on both sides, and conclude that the quantity $$\eta=\frac{x}{\sqrt{\kappa t}}$$ is dimensionless. From here, it seems every source on the internet says "since $\eta$ is dimensionless, it says something about the system, so try a solution of the form ...", without deriving anything. Is there any way to get some information out of this without guessing? My question in particular was about the following: suppose $\theta=\alpha t^{-p}F(x,t)$, $\alpha$ is a constant; then show $F(x,t)$ is a function of $\eta$ only. I feel like the step that is skipped when websites guess the solution is the step that I am missing to answer my question.