Which characters have full unrestricted access to lower-leveled content?

Since I flew through all my lower levels, straight into 4th Job, avoiding/missing PQs and with level capped content coming (I'm looking at you Kerning Square) I decided to remake a low-leveled content/PQ mule so I can re-experience these things and grab the equips I missed but want to collect (Slime Shoes, Pirate Hat, Altaire Earrings, etc.)

Originally I decided to make a Blaze Wizard because they level quickly, however I think I can remember some PQ restrictions placed on them, are Adventurers the only characters that have unrestricted access to all PQs, if so, what class would be best suited for this purpose (PQer)?

Important PQs include:
Ariant
Henesys
Kerning
Monster Cleanup
Ludibrium
Ludibrium Maze
Monster Carnival 1 & 2
Ellin Forest
Magatia
Pyramid
Pirate
Orbis
Balrog

I'd say either a dexless assassin or a lukless magician. The assassin will help you in Carnival PQ and Magicians are good in a variety of situations.

If the goal is mass PQing, I'd try a Lukless Cleric. Everyone likes Heal.

I'm gonna have to recommend any character that mobs and has a pretty good single target skill. I'd say some kind of warrior, a spearman is probably best, an I/L wizard, or one of the pirates, brawler being the best. Mobbing is good for a lot of quests, including PQs, while single target damage is good for bossing, and I'd say mobility is a pretty important factor, it helps you get around easier.

KoC and Aran do not have access to Ellin Forest PQ (or CWKPQ, but that has no upper limit, lol).
Seems like people always need mages for Magatia PQ (in Mardia at least) (require max tele).

KoC and Aran do not have access to Ellin Forest PQ (or CWKPQ, but that has no upper limit, lol).
Seems like people always need mages for Magatia PQ (in Mardia at least) (require max tele).

this 99 % of the time, if you are missing one role in MPQ, its a mage.