Ah, that explains it. I rarely use iMessage for anything but quick texts to people who aren't in one of my other messaging apps, so I hadn't looked too closely at it until I checked it out to compare the iOS flow to the shit-show of an Android one I'd just dealt with.

Checking it again, I see now that the grey app-store-stack icon is toggling the horizontal app list, so it's not that it was bringing up the suggestion list before, but that it was turning off the app list, revealing the suggestion list. Of course it doesn't in any way visually represent a toggle, as its appearance remains the same whether it's "on" or "off", and it's not visually tied to the app list, so the connection is rather metaphysical. Sigh. But, at least now I know how it works.

I generally like Apple design well enough but IMO they derailed way off the usability/discoverability tracks back in iOS7 and have yet to fully recover—it's just that almost everything else is even worse.