Im playing on TLE-server and I really cannot choose between the Conjurer or the Illusionist. With the COnjy I really like the pets and the solo abilities but when it comes to grouping I really don't feel im that attractive, I really only give dps like so many other classes also can do. When it comes to the Illu I feel I can solo quite well but ofc not as well as the Conjy on the other hand I hand out nice buffs that the group wants. Would you rather pick a conjy or an Illu for your group? What class do you find has it easier to find a group?
There may be reasons you're not being invited to group that have nada to do with the character. Conjy has so many different options available. It's a rare group I see without one, tbh. Frankly, not a lot of grouping going on these days, so are you willing to give up soloability to just wait on getting groups? I can't speak to the TLE server and grouping opportunities. I've been on my conjy when all that was needed was shards. But honestly, play what makes you happy and that you'll be happy to log on and play--even if it's by yourself.
I have one of each, though my Conji is a bit older. (I don't Group) I enjoy both, but I like elemental pets more than the Illu doppelganger, mainly because there is more variety appearance wise. Conji pets also have different functions depending on which you are using, whereas the Illu Persona is basically a one trick pony. But, as others have said already, Play as you want and enjoy things your own way. This isn't Follow the Leader, after all.