Getting the Primal/Trial spells sucked. That's the simplest way I can describe it. As OP noted, you basically need to bribe people or hope someone is very bored to run the same encounter dozens of times with no promise you're any closer to actually learning what you want. Sometimes BLUs do the "trade" thing where they'll run a higher level job for them, then switch, repeating until spells are learned. In some cases, this is feasible, but it also sorta feels like you're cheesing the system since it's not really what I'd call "playing the job" when all you're doing is mooching.

Personally, I'd suggest the following:
1) Primal/Trial monsters have Totems added to their loot pool that will teach their spell if not learned normally. These could only drop if a BLU is present.
2) The consumables Primals drop can be traded for a Totem, say something like 5 per.
3) Better improve chances with multiple BLUs in the party, with 4+ being 100%.
4) Add Carnivale versions of the various Primal/Trial/Dungeon Boss encounters that aren't super gimmicky. Their main purpose should be learning.
5) Award Tomes to helpers in instances when a BLU learns a spell. Further link an achievement to this process that could reward something like a mount. Ultros, maybe?

The ability to queue if meeting specific spell combos or doing things with Squadrons are certainly issues, too. I'd also like a healing spell that isn't White Wind, but I imagine that's in the update pipe. A reraise with a decent recast wouldn't be out of the question, either. I do agree that Garuda and Leviathan are problematic if skipping mechanics and not bringing a healer. I'd lump Ifrit in here too, but fortunately you can solo Eruption on normal. Having to wait so long for Shiva to go to bow mode is also not something you can expect randos to be patient with.

Honestly, there's not really anything about BLU that I see that would mandate it remain a limited job in the future. If they don't want us petrifying or dooming something, then by all means, just make those mobs immune like so many seem to be with paralyze.