Non-Hausdorffness shows up in several contexts when dealing with Lie groupoids: the integration (Lie's 3rd Theorem) for Lie algebroids to Lie groupoids will typically produce a non-Hausdorff one, if it works at all. So there seems to be a large part of oid-geometry involved with non-Hausdorff manifolds.

However, to abandon second countability is a more serious step, at least in my eyes. It is not so much the existence of partition of unities (which requires paracompactness) but the second countability itself which is extremely useful. Consider the following statement:

A bijective immersion is a diffeomorphism

This is a theorem in differential geometry which you definitely want to be true and it relies directly (well, a bit hidden) on second countability. To see that it fails right on the nose if you drop second countability, consider a manifold $M$ in the usual sense of positive dimension and $M_{\mathrm{discrete}}$ as a zero-dimensional manifold with uncountably many connected components, each of which is paracompact (it's a point...) Then $\mathrm{id}\colon M_{\mathrm{discrete}} \longrightarrow M$...

Now why is this theorem important: Lie theory depends strongly on it. Any group would be a zero dimensional Lie group otherwise. In particular, the manifold structure of a Lie group would not be determined by the group structure. This would also imply that a transitive smooth group action of a Lie group on a manifold is not the same thing as a homogeneous space $G/H$ and many more problems...

So before asking why one should abandon a property, it might be good to understand what it is good for. For Hausdorffness the situation seems to be very different to second countability.