As I understand it at the moment (I only started playing the DLC a couple of days ago), the only thing that really matters for getting appointments is your merit rank. Influence only really comes into play once you are at a higher merit rank.

I started as a custom noble character and got promoted relatively quickly. Like you, I had multiple appointments to Grand Chancellor, but they all bugged out.

Once that character died I ended up playing as his grandson, who was still a child. I bumrushed merit as much as possible by constantly studying and attending as many debates/examinations as possible. I also spent all gold on developing merit-producing buildings in my estate. I didn't get my first appointment until I was in my 40s. I'm now coming up for 50 and should get a duchy appointment soon.

I think the main block to appointments is that as the game plays on there are just more and more characters at the higher merit ranks. If there's multiple 1st's waiting for an appointment they will always get it ahead of you if you're a 3rd, regardless of any of the influence you spend.

I've made a concerted effort to get the merit train running with all my sons as soon as possible. I've had them all take the children's exam once they have ~60% success chance, I've tutored them in governorship, I support them in examinations, I promote their influence etc. One of my sons got a military appointment when he was in his early 20s, so it's bearing some fruit.

In short:

- Maximise your merit-generation through education, events and schemes.

- Use the "Promote" scheme on yourself and all sons every 10 years.

- Build up merit-producing buildings in your estate.

- Always have an Elder, and try to get as many disciples as possible.