Competition, whether PvE or PvP, is one of the least fun things about MMOs and won't improve anything in FFXIV. MMO competition is grinding, which is why it's not fun and can't be fixed.
The way to win competitions in MMOs is outfarm your opponents.
For PvE, run more dungeons, farm more materials, craft more stuff, or spend more time grinding than anyone else.
For PvP, grind gear to farm players who have grinded less and have little chance to win.
Farming competitions people will not create a sense of community.
Giving relics horizontal progression is also a terrible idea. If it's BiS, then raiders will have to farm it in order to raid. If it's not, then everyone will ignore it.