A closed form means that there is a defined number of solutions that solve the problem. https://en.wikipedia.org/wiki/Closed-form_expression
The issue with two Gaussian variables is that they give a circular joint probability for which rotation is arbitrary, so there is no single solution.
An ellipse can be considered simply a circle where one axis is scaled differently, but this scaling can be applied to any arbitrary rotation of the unit vector relationship between them.
Basically you are looking at the flip side of the issue covered in several questions on Independent Component Analysis.
What is mean by the non-gaussianity in the independent component analysis(ICA)?
The Independence in Independent Component Analysis - Intuitive Explanation
Numerical estimation is used to provide a way to break the impasse of an open solution. We can't identify finite (or manageable) unique solutions analytically therefore we apply some decision rubrics to allow us to move forward. Its a compromise that accepts we would waste eternity trying to find the true solution, but we need something we can act on in a finite period of time.
What those rules look like depends on what matters to the analyst and likely some will be completely arbitrary. The key is that we accept a certain risk that our chosen solution is sub-optimal to allow us not to become frozen in indecision.