"Paladin Healing has no problem keeping up with Priest healing."

That is a completely misleading statement. Paladins are not competitive with the other healing classes in AoE raid damage situtations, but are still undisputed in raw single (or two) target HPS.

AoE healing that Priests/Shamans were better at than Paladins continues to be a place where Paladins are outmatched, but now Druids are also good at it. Raw single target healing is where Paladins used to be good, and they still are the best.

Healing is one of the only areas that didn't change much with regards to the consolidation of the classes. Each class is still excels at particular situations, and for the most part the same particular situations they used to excel at.

The spirit healers (priest/druids) are pretty well suited for any healing role now (druids used to struggle with bomb healing and AoE, but both issues have been fixed).

Shamans are in my opinion the premier non-tank raid healers due to the continued excellence of chain heal coupled with the new instant cast riptide for rapid spot healing. They are better at tank healing than they were, but would very rarely be used primarily for that role unless none of the other healing classes were present due to the massive mana inefficiency of healing wave.

Paladins continue to be the best class at pumping out massive heals reliably and quickly. Beacon of Light does not bring them up to par with healers with a real AoE heal, but does make them extremely effective at either healing two tanks or a single tank while sniping quick spot heals on single targets. Holy Shock+Infusion of Light have hugely improved the problem of healing on the move for paladins (particularly with the recent chance to proc instant cast flash of lights).