Stiff systems of ODEs such as the van der Pol oscillator and Robertson's chemical reaction do not have exact solutions (as far as I know).
Are there stiff systems of ODEs with known exact solutions, perhaps other than simple systems such as $y' = Ay$, where $A$ is some diagonal matrix whose diagonal elements vary significantly?
Using the wonderful notes "STUDENT VERSION, Stiff Differential Equations" by Kurt Bryan
Example 1
As you noted, the simplest example is to just create a diagonal system
$$y_1(t) = \lambda_1 y_1(t) \\ y_2(t) = λ_2 y_2(t)$$
The exact solution is given by
$$y_1(t) = y_1(0)e^{\lambda_1 t}, y_2(t) = y_2(0)e^{\lambda_2 t}$$
by choosing $\lambda_1, \lambda_2 <0$, for example $\lambda_1 = -1, \lambda_2 = -100$, we get stiff behavior.
Example 2
Can we find a constant matrix with the same eigenvalues as the previous problem? Yes, choose
$$y' = A y = \begin{bmatrix} -56 & 55 \\ 44 & -45 \end{bmatrix}$$
The exact solution is given by
$$y(t) = c_1 e^{-t} \begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 e^{-100t} \begin{bmatrix} 1 \\ -0.8 \end{bmatrix}$$
Example 3
From the amazing book "Numerical Methods for Ordinary Differential Equations" by J. C. Butcher, a mildly stiff system is given by
$$\begin{align} y_1'(x) &= −16y_1 + 12y_2 + 16 \cos(x) − 13 \sin(x), y_1(0) = 1 \\ y_2'(x) &= 12y_1 − 9y_2 − 11 \cos(x) + 9 \sin(x), y_2(0) = 0 \end{align}$$
The exact solution is
$$y_1(x) = \cos(x), y_2(x) = \sin(x)$$
Example 4
You can find more examples of stiff systems with closed-form solutions at link, including an RC-circuit (for a real system using an electric circuit).
There are many solvers for stiff systems.