I don't think that trying to solve the Schrödinger equation itself is particularly useful. The SE is a partial differential equation, and there's a whole logic of differential equations and boundary conditions, etc. that provides context for the SE. If you're serious about trying to understand quantum mechanics, I think the concept of Hilbert space/abstract vector spaces/linear algebra in general is a bigger conceptual shift than just being able to solve the particle in a box in function space. It's also just a really useful set of concepts that makes learning things like optimization, coordinate/fourier transforms, etc. easier/more intuitive.

Until I had the wave function explained to me as some vector in a high dimensional space that we could map into x-space or p-space or Lz-space I don't think I really had a good grasp on quantum mechanics. This is anecdote not data, your mileage may vary.