So I've played a decent amount of Tamer, Wiz, Witch, and Nova, and I think something that they all have in common is that while they have pets, they're more or less just an extra added-on-top kind of thing. These classes more or less play the same with or without the pet out.
So, I think i good idea for a new class is one that's truly based around the usage of a pet/summon, where instead of it doing some added effect on an attack, it itself should be the attack. The class shouldn't have a weapon to hit things with or cast spells with and instead your basic attacks and all abilities should use the pet/summon, and the mainhand weapon should be something used to conjure and control the pet/summon, like a wand or a flute or something. This would probably also require the pet to be more built in to the class, so it should be out at all times from level 1, rather than having a spell to summon it and it having a 30-60 minute duration. It would also add to the class fantasy to have the pet/summon hang out with you and follow you around everywhere.
I think that one of the reasons the current pets/summons don't have infinite duration is probably something to do with not being able to have it out attacking things for you while afk, so maybe this pet won't have any attacks it goes and does on its own, and it only attacks when you use an ability/basic attack.

Aaaah, I see what you mean... But for the Tamer, what was interesting for me in BDO was precisely the fact that our skills were activating Heilang's skills!

In fact, for a real "pet master" feeling, I think it would be best to have a list of Heilang's skills and being able to choose which tamer skill will activate each of them, and the delay.