The issue with FATEs is currently reward vs time.

At the moment, FATEs give a little bit of experience (About 100k-150k in SB, going up to 200k-230k with Twist of Fate bonus), insignificant amounts of Gil and insignificant amounts of GC seals (Which are an outdated currency and have been since ARR >.>)

Meanwhile, you have Dungeons and PotD/HoH that give a ton of experience (Like 1-3 million in SB), give a small amount of Gil, can give a TON of GC seals (From trading in loot), and also can give Poetics (Largely outdated currency, but at least semi-relevant for ARR/HW relic collectors)

Meaning, that in order to similar experience to doing 1 duty, you have to complete 5-20 FATEs (Depending on if you're lucky with Twist of Fate bonuses, most likely you'll see 1-2 within the time it takes to clear out a bunch so it's more like 10-20 average) and if you want Gil/GC seals/Poetics it is just hands down better to never do a FATE over a duty.

Thus the time to complete a FATE should be about 10-20x faster than a duty. It is not. A duty takes like 15-20 minutes. FATEs take about 5 minutes (Maybe longer if you are solo)

So basically, the easy way to make FATEs relevant, is to buff their rewards to being actually worth the time it takes to do them.

Flat out double their base experience gain (So 200k-300k in SB so about 400k-600k for Twist of Fate bonus). Increase their gil reward by 10x (So you get 2k gil per FATE with gold reward). Slap in some semi-relevant currencies such as Poetics for ARR/HW, Mendacity for 60-68 and Genesis for the level 69 FATEs so that FATE farmers can then purchase tome gear when they reach relevant levels (Potentially add in a few more sets to purchase at like level 30, 55 and 65 to make currency while leveling relevant)

Something like that would go a long way to making FATE farming somewhat relevant, even if it's just while leveling.

If you wanted to promote Trains too, put in daily bonuses in specific zones. Each day will be a new zone that gets like 50% more rewards and maybe some Genesis tomes irregardless of level range. That way people will funnel into the same zone to farm FATEs.

In addition, you can increase the rewards for completing the Raid FATEs. For example, put a weekly raid page reward for doing a specific Raid FATE. So that Raid FATEs are also worth doing.