I don't really have experience with reconnecting with friends, so my answer won't really be complete. However, I am non-binary and I also do have experience with unexpectedly reaching out to someone in order to bring them joy.
I write about such a reach out in another answer here so my post will partially be based on that (and another reach-out that I also did in the past).
As I said, I don't have experience with resuming friendship, so instead, I'm gonna talk about bringing happiness to this person. Hopefully, this will be enough to allow you to resume the friendship.
As a trans/non-binary person, here is what brings joy to me regarding this topic:
When someone use my choosen name even though I didn't explicitly asked them to do that.
When someone use my chosen pronouns even if I was (supposedly) not around to see it.
You have to know that, when you are transgender, a lot of people will use your chosen pronouns when you are here but, as soon as you aren't, they will revert back to pronouns you no longer use.
Those two things could appear very small, but it always makes me unexpectedly and amazingly happy.
When people do that, it shows me that they accept me fully and it's an incredible feeling.
Saying to someone "I accept you as transgender" can be nice but isn't necessarily enough. A lot of people say that but don't really mean it and/or still have really awful ideas about what being transgender actually means.
When I meet new people, I'm often on my guard, but it's really reassuring if I see that they arlready know a lot about the trans community (then, I might feel safe enough to tell them that I'm transgender).
So, if you want your friend to feel safe with you as a transgender person, you have to show/prove them that knowledge. Only then, you will be able to support them (because, from experience, the support of someone you do not trust isn't really meaningful).
One quick way I found to differentiate someone who knows about the trans community and someone who doesn't is to check if they know the transgender flag.
So, when reaching out to your friend, you may want to show them a nice drawing involving the transgender flag (or other transgender symbols).
For example, you could say:
Hey, I saw this amazing drawing the other day and it made me think about you.
(If your friend is more into "cuteness" than "badass", you could use a drawing like this one instead).
4: So, by doing that, I think you will accomplish your latest goal: "Make it clear that their transition is not the reason for [your] silence". I know that, if I meet someone with enough knowledge of the trans community, I probably won't expect them to be transphobic. In fact, I am way more likely to believe the opposite and trust them more quickly.
2: It will also bring you closer to accomplish your second goal: "Be supportive of them". Since they will know now that they can trust you, they will accept your support more easily (at least, that's what I tend to do).
1: Also, it might also help resume your friendship (goal number one). At least, it will probably make them happy that you thought about them enough to 1) send them a message and 2) send them a nice picture with it.
I know that I'm always happy when I learn that someone is thinking (positively) about me and I also know that, to me, receiving a picture is like receiving a gift. If your friend is at least a bit like me, they will probably appreciate the attention too.
3: As for the apologizing part (goal number three), I don't have any advice except one: Do apologize. People don't like it when you hurt their feeling and don't apologize. It makes them feel like you don't care about their feelings at all, so apologizing is always something one should be prepared to do.
Note: a lot of "newbie mistake" that people make when talking to someone transgender is to ask them about their genitals or other physical transformation they may have done/may want to do.
Don't do that. Those are very personal questions and you should never ask them to a transgender person unless 1) you are very close or 2) they bring it up themselves.