Its completely my personal opinion but I think it'd be neat if triggers could drop from other 24/7 content in the game in addition to the current system, perhaps drops of specific triggers could be increased or decided based upon the region/field the quest takes place in. (for example, you could get triggers for Elder from Naberius more often, Mining Base from Lilipa more often etc.) and also the difficulty/type of quest (SH AQ's could have a further boosted drop rate for example), this would allow people to play both new and old EQs on their own time.
Of course ideally all of these EQs should drop stuff that is somewhat relevant, like how when 14* drops were first being added, some were able to drop off of older bosses like loser and PD for example. an alternative solution would be to just have all the EQs drop some general "EQ currency" like the crests we've been getting that you can use to purchase relevant gear and stuff
Getting players would definitely be the hardest part to tackle, the only thing I could think of other than keeping the current trigger system would be to make it more like extreme passes where everyone would need a trigger to enter (drop rates for them would likely have to be higher then), and it would just matchmake you with other people who have triggers. I think making stuff like EQ boss health scale with player count would further help with the issue of not finding a full party of people, since you could start with a lower player count and as long as everyone pulls their own weight you can still clear it.