My guess would be that you can still earn the achievments/mounts if you have a title - but thats really my guess, nothing official.
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Also when you play through the Heavensward and Stormblood main story quest lines you will gain at least four new mounts by default as well (three for HW and one for SB). Additionally, the old 'Veteran reward' system had mounts as several of it's rewards for long term subscription to the game (ahriman at, 90 days I think it was, and behemoth at.... 180?). But SE changed the Veteran System with Stormblood's launch so those mounts are gained differently now (by unlocking achievements and spending the points you gain for 'achievement certificates' which are redeemed with Jonathas in Old Gridania).
Having your character marry another player's character (the Ceremony of Eternal Bonding) also gains you a mount (Ceremony Chocobo) so that is another one to get. ;)
Also sadly some mounts are now unattainable, with at least one (Legacy Chocobo) will be never be offered again sadly. But you can still get the mounts that are available (also there are quite a few found for sale on the Mogstation). :)
This site lists all the mounts currently released in the game.
The primal ponies are easy to get (sans Leviathan - he is easy but it takes f o r e v e r).
But if you want a cool flying mount, once you hit 60 you can do Ravana Extreme real easy and quick for a cool glowing red falcon. There is another extreme primal that is pants-on-head easy but I do not know if you have seen the said primal yet so I will withhold the name for now - but you get a light blue falcon that has bubbles from it!
I find these quite easy to get:
Wedding chocobo and other Mog Station mounts
Collector's Edition mounts
Recruit-a-friend mounts / mounts bought with Gold Chocobo Feathers
The cheapest Gold Saucer mounts
Mount bought with Centurio Seals (level 50+ hunts)
Tradeable mounts on the MB
Palace of the Dead mount
Level 50 primal mounts farmed unsynced (level 60 primals have a worse drop rate)
Level 30 conjurer quest mount
There is a mount from commendations.
If you'll be farming extreme primals, do that on a tank so you get credit towards tank mounts too. Garuda Ex is probably less than 1 minute with lv70 dps unsynched, you need 200 runs per job. SB tank mounts are better farmed in Omega Deltascape 1 which is around 5 minutes, or Deltascape 2-4 at around 8 minutes.
Oh yeah, forgot about that one. Only300500 commendations needed for it. And then another at 3k(I believe).
Edit: Yup, one at 3,000 commendations. The Parade Chocobo Whistle will be yours(only 596 more until it is mine)
Woops, you are correct. My bad.
Good news! I just got my first easy mount! The Zu mount with the Zu Horn! Bought in the market place.