Laplacian of Gaussian Approximation and Gaussian Blur as the Solution of Heat Equation
I'm not sure I fully understood what's the issue you're having.
Yet I will show a simple property of the Gaussian filter which might make things clearer.

For simplicity, I will use 1D Signal.
Yet it is easy to extend it into 2D.

The Problem

For $ u \in \mathbb{R} $ (1D, unbounded domain), show that a Gaussian convolution with the initial condition solves the linear diffusion equation:

$$ {u}_{t} = {u}_{xx}, \quad u \left( t = 0; x \right) = f \left( x \right) $$

That is:

$$ u \left( t; x \right) = f \left( x \right) \ast {g}_{ \sigma \left( t \right)} \left( x \right) $$

Where $ \ast $ denotes convolution.

Find the function $ F \left( \cdot \right) $ relating the
evolution time $ t $ to the Gaussian standard deviation $ \sigma \: : \: \sigma = F \left( t \right) $.

Solution

Looking at:

$$ \frac{\partial u \left( x, t \right ) }{\partial t } = \frac{\partial^2 u \left( x, t \right ) }{\partial x^2} , \; u \left( x, t = 0 \right) = f \left( x \right) $$

Defining $ U \left( s, t \right) $ as the Fourier Transform of $ u \left( x, t \right) $ with respect to the parameter $ x $.
This leads to:

$$ \begin{align}
\mathfrak{F} \left( \frac{\partial u }{\partial t} \right) &amp; =  \frac{\partial U \left( s, t \right ) }{\partial t}  \\
\mathfrak{F} \left( \frac{\partial^2 u}{\partial x^2} \right) &amp; = -4 {\pi}^{2} {s}^{2} U \left( s, t \right ) \\
\mathfrak{F} \left( f \left( x \right) \right) &amp; =  F \left( s \right)
\end{align} $$

Hence the equation in the Fourier domain is given by:

$$ \frac{\partial U \left( s, t \right ) }{\partial t} = -4 {\pi}^{2} {s}^{2} U \left( s, t \right ) , \; U \left( s, t = 0 \right)  = F \left( s \right) $$

The solution of this Linear Ordinary Differential Equation is given by:

$$ U \left( s, t \right) = F \left( s \right) {e}^{-4 {\pi}^{2} {S}^{2} t } $$

Hence the solution in the time domain is given by (Using the identity $ \mathfrak{F} \left( {e}^{- \alpha {x}^{2}} \right) = \sqrt{\frac{\pi}{\alpha}} {e}^{- \frac{{\left( \pi f \right)}^{2}}{\alpha}} $):

$$ u \left( x, t \right) = \frac{1}{\sqrt{2 \pi 2t}} {e}^{- \frac{{x}^{2}}{4 t}} = \frac{1}{\sqrt{2 \pi {\sigma \left( t \right)}^{2}}} {e}^{- \frac{{x}^{2}}{2 {\sigma \left( t \right)}^{2}}} $$

Comparing it to the classic Gaussian Function yields that $ \sigma \left( t \right) = \sqrt{\left( 2 t \right)} $.

Remark
This is a solution of mine to a problem I was given in a course named Variational Methods in Image Processing.