Ok so I have lurked for awhile now, and I haven't seen anyone talk to much about the Ryshadium and I find them very interesting so here goes.
I think that there is a lot more going on with the Ryshadium than we know at this point. Wit/Hoid who knows a whole lot about what's going on in the cosmere and on Roshar has this to say,
When Hoid is surprised by something... well we should take a closer look, and expect some awesomeness.
Also they Ryshadium know when their master is in danger. See Gallent charging through the entire Parshendi army to try to save Dalinar in WOK. I also recall the time when Adolins friend after the battle on the plains refuses to drink with him back in the war camps because Adolin is unpopular. Sureblood attempts to confort Adolin:
There seems to be a direct emotional link between the Ryshadium and their rider. I perosonally think there is a link very similar to the Nahel bond, but obviosly not with the same powers involved, that goes on between rider and Ryshadium. I suspect this will not play a major plot centric role in the series, but it is part of Brandon's great world building to have these intelligent BONDED horses on Roshar. We know that only the "worthy" can bond with the Ryshadium. Sounds a lot like the Nahel bond to me.
These, plus the quote that has been mentioned about the only the radients having Ryshadium in the beginning, really makes me think there is something going on.
Personnal side theory: Sureblood's death and loss of the bond will leave cracks in Adolin's soul and open him to a possible Spren bond.