Corrugated might provide a little more protection than just the record sleeves, but I'm thinking that some 1/4" plywood would be better.... that would certainly keep the upper corners from bending, but then you still might have the problem of the records sagging below the protection of the boards...
Maybe another skinny piece of ply on the bottom, hinged to the uprights?
Where are you getting vinyl? I've been playing more, recently, and was thinking that I needed to go peruse some shops, which I haven't really done since moving back here. There's the place on Nicollet and 26th... and the place on Portland north of the crosstown... the Electric Fetus... where else is good?
And that video doesn't seem too far off from what I see on Strava....