I've almost uniformly studied the homological algebraic aspects before I got around to studying the corresponding results from algebraic topology. It did get somewhat artificial at points - specifically triangulated categories make a lot more sense once you've seen Serre fibrations than before you do.

I felt quite well motivated by the approaches I encountered though; with the study of Ext and Tor to divine interesting ring properties taking the forefront in homological algebra, with a side dish of approximating modules by things that are free everywhere that matters, but sacrifice degree concentration to achieve it.

My personal feeling is that it probably depends to a large extent on whether whoever is teaching the material wants to teach homological algebra or algebraic topology: if you're happier thinking about topology, then homological algebra will feel desolate and artificial almost no matter what you do about it; while if you are genuinely interested in homological algebra on its own, it's much easier to sprinkle in the off-ramps as you go, pointing out where certain concepts have roots outside the current area, and how to get more information about the roots.