Am I missing something here?

[XenoEvil]

I finish the game on moderate difficulty with my characters being in the low 50s, but there are quest that I have encounter where enemies and bosses are in the mid and high 60s, specifically areas where the other lights from lenegis fell on dahnan. I have been fighting every enemies whenever I get into a dungeon and rarely do I ever avoid any battles, doing as many side quest whenever possible, etc. Is the game expecting players to brute force through these quest being massively under level? WTF?

[Simon]

You have to farm some EXP basically. Just eat exp boosting food and farm a little bit of level at the final dungeon or something like that. If you do the Earth Temple quest, you can access a bonus dungeon, where you can level up much faster

[Ivy in my Heart]

Good news, you're not missing anything, low to mid 50s is a perfectly normal level to finish the main quest at. :)
Even better news, there's practically no grinding necessary to take on the bonus quests and EX dungeon in post-game.

Your level is really enough to tackle the higher-level sidequests (Phantom Flower of Nevira, Beyond the Grave, Untamable Rage, Super Zeugle... etc.) and survive them with a couple of life bottles. XP-enhancing food and 2 happy bottles before the fight help you gain levels even faster.

After all those quests, you will be able to take on the first boss of the Otherwordly Visitors quest, aka the EX dungeon. Just normally fighting through all the levels of this quest will get you to at least level 90, after which you can fight the quest's end boss 4-5 more times to reach level 100 if you're going for the achievement.

Many people recommend a really small (like 20-ish minutes) grinding session at the bottom of the tower of the wind spirit at Mount Dhiara. There's a very easy mob of bird enemies there. With 2 happy bottles per fight + XP-enhancing food it should only take a few battles to get to level 62 or so, which is a great level to start the EX dungeon.

[Lyff]

Nah the truth is you missed both XP bonus 'artifact', one give +20% XP & the other one give +80% for a total of +100%, you usualy get both near lvl 30-40 by completing quest, forgot which one tho. Also the difficulty in which you play does a lot. Playing 'Unknow' I was near lvl 70 at the end of the main quest without any farming out of quest, tho I'd finished all available quest before finishing main quest, so with post-game content I ended up 94 when finishing everything.