Dead shardblades act like fabrials too. The bonded gemstone allows them to be summoned; though it takes 10 heartbeats unlike a 'live-spren' shardblade, and dismissed. Dead shardblades also require a bonding time period. The live blade is available as soon as the WORDS are spoken. Perhaps shardplate acts more like a fabrial because of the lost connection with the KR. Non-KR shard bearers have to feed the plate storm light for it to rebuild itself. Does a KR have to do that and if so, is it done in the same way? Probably not in the same manner.
I recall Dalinar saying that the shardplate worn by the KR shined were as current shardplate is usually painted. It's apparent that there is a difference in how a KR is able to bond, interact with and use Shards compared to non-KR.
I'm waiting for Kaladin to speak ALL of the WORDS and have his shardplate appear to help him defeat Odium's champion; or something as equally dramatic as Syl appearing in his had as a 'Shard-weapon' to help him defeat Seth. However the appearing happens, I don't see how Syl (or other Nahel bonded spren) could possibly NOT be involved.