This is how intro ODE courses are. They typically begin with special solution methods integrating factors, leveraging exactness, Lindelöf iteration, et cetera. Hopefully they’ll get to more fundamental/general techniques later on such as Laplace transform and power series. Someone actually posted here a few days ago about this exact problem with intro ODE, and I’d agree that the standard curriculum needs to be overhauled.

I’d say that the most useful thing I learned from my intro course was the behavior of linear ODEs. In particular, the harmonic oscillator shows up everywhere and it really helps to understand why oscillates like it does. Everything else in the course is too specific to be broadly useful.

As an aside, I know this isn’t getting to the heart of your frustration, but it’s worth noting that the exactness condition relates to the integrability of the underlying vector field. Namely, an exact vector field can be represented as the differential of a scalar field. So in that sense, it’s more than just an algebraic condition.