Any shortcuts to understanding the properties of the Riemannian manifolds which are used in the books on algebraic topology
Bröcker and Jänich's "Introduction to Differential Topology" is admirably short and nicely written, and you may find that it contains everything that you need.
The proof of the existence of good covers is contained in Bott & Tu on pages 42-43, though that proof does refer out to Spivak (see below).
In general, if your main goal is to study (algebraic) topology of manifolds, you probably don't need to know much about metrics and connections and that sort of thing that typical differential geometry books spend a lot of time on. It sounds like what you really need is some material on differential topology. I don't know how much they'll cover of the specific things you're looking for, but here are a few suggestions to check out:
Topology and Geometry by Glen Bredon. This might be a particularly good book for you as it really combines the two topics pretty well.
Michael Spivak's A Comprehensive Introduction to Differential Geometry, Vol. 1. Despite the title, the first volume is more about differential topology than geometry. Also this is what Bott and Tu cite for their key fact needed for the existence of good covers.
Differential Topology by Guillemin and Pollack - this is a very readable introduction.
Introduction to Smooth Manifolds by John M. Lee - this is oriented a bit more toward geometry but you can find a lot in it.
I'd recommend skimming through these (and others) to find one that suits you and has the kinds of things you're looking for.
There's a short book by Milnor: "Topology from a differentiable viewpoint" (here is one link https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf where you can find it).