Why is the Sahara so much larger than the other horse latitude deserts?
I think the size of the continent is a big contributor.  If there's very little cloud cover, and little rain, the air just keeps drying out the ground.  Try finding a good map of surface or lower-level winds, and just compare how long it takes an air parcel to go from over ocean to the Sahara.  Sometimes the winds will come straight from the oceans, but often they come from over land: the Near East and Asia.

But yes, all your examples end at mountain ranges.  Or rather, start there.  At the horse latitudes and on the equator-ward edges of the subtropical ridges, winds mostly blow east to west, not west to east.  A number of these deserts are thus in rain shadows of a sort (https://en.wikipedia.org/wiki/Rain_shadow).  This kinda ties in with the above: Since the wind blows east to west, the air that blows over the Sahara hasn't just been over Africa but also much of Asia, drying out along the way.

My explanation for Australia would be this.  It doesn't have major mountain chains, so it's not in a rain shadow.  But it doesn't have mountains, so there's nothing to initiate orographic precipitation.  It's convective precipitation or nothing, once you get inland.  And since the ground is dry, convective precipitation doesn't have much of a water source, so it doesn't rain much and the ground stays dry.  Someone with more background on Australia should chime in on that, though.