The intuition is correct, although I would amend your statement that the smallest eigenvalue of the operator and its matrix discretizations is always strictly positive and hence never degenerated.
Putting it another way, this is why in infinite-dimensional spaces, it is not sufficient for a (self-adjoint) operator $A:V\to V^*$ to satisfy $\langle v,Av\rangle >0$ for every $v\in V$ but you need to have $\langle v,Av\rangle \geq c\|v\|^2$ for some $c>0$ independent of $v$.
But note that if what you want to show that a matrix $M$ is invertible (which is what the point of the Lax-Milgram is), positive-definiteness is not necessary, only sufficient -- you can have invertible matrices that are not positive definite. All that is required is that $M$ is injective and surjective.
The corresponding theorem for infinite-dimensional operators is the Banach-Necas-Babuska Theorem (see, e.g., Theorem 8.1 in my lecture notes), which replaces the Lax-Milgram Theorem when studying Petrov-Galerkin methods such as DG. Again, you need to make the assumption "quantitative" in infinite dimensions: instead of surjectivity, you need the so-called inf-sup condition.