Oh, the orbit matters... but the orbit is EXTREMELY predictable even its wobble and orientation. It might, perhaps tip the scale during the (likely ongoing) pleistocene (Where 90% of our time is spent in ice age with 10% warm snaps that should have already ended by now, contrary to spiking upwards instead) but ebbs on a timeline that should have had the pleistocene happening essentially since beginning of observable time (which it has not.) So, it's a factor, but not a decisive one. Continental arrangements and landmasses propensity for temperature extremity vs. oceanic propensity for temperature moderation and long-distance transport matter far more (even than tilt, given measuring the southern hemisphere vs. northern.) And yet, in spite of the fact that the continents and oceanic currents are still in the same messy tangle they have been for the entirety of the multi-million year pleistocene, these beasts didn't go extinct during an of the previous warm-snaps... just the one we arrived in... and now that we should be quickly descending into ice age, instead we're headed the other way.
This article is of interest, but it is not argument against anthropogenic extinctions or climate change.