This is an illustration of why I think the existing system's terrible (even if the alternative might be worse); I have little doubt there are helpful mentors on your server, but unless you bump into them, how do they know to reach out in the first place? If you bump into someone in a dungeon and they offer help and strike up a conversation, that's easy. But how does a mentor know that this particular sprout out of the 20 around the aetheryte needs help? Just /tell'ing doesn't seem great either; I know if I were a sprout and every time I was around an aetheryte I got three different mentors /tell'ing me "Hey, do you need any help?" Sure, they'd probably have the best of intentions, but I'd probably quickly figure out how to remove that sprout icon. But if it wouldn't be ideal for mentors to just spam the sprouts, how does a sprout seek out help on their own initiative? Even assuming they know to find a mentor in the first place, they may not know how to use search tools to do so. And even if they do that, supposedly not all mentors are receptive to such requests (based on the "Mentor system is broken!" type threads).
What might work better, now that I'm poking at my brain to see if ideas fall out, would be to make the Newbie Network work more like FC applications? It's possible for someone to apply to an FC without having to talk to someone, and FC officers can look at pending applications to approve/deny them. Make it so a sprout has the option to say "I want to join the Newbie Network" and then mentors could look at a list of applications (and filter by who's online right then). That way they could reach out via /tell to make sure it's a real player and not a bot, and just approve the application. Or (if the player status system were made less opaque and more discoverable) a "Looking for Help/Advice" status icon a sprout—or even an advanced player—could set on themselves, and then a mentor (or even just another experienced player) could reach out to them upon seeing that.
Beyond that, I'm sorry you've had trouble making ties in game; you're right it's easier to stick around when you do. I bailed on the game ages ago around Heavensward, and made a very very brief attempt to return around Stormblood (and never even finished my trek into Heavensward). This time when I came back, I found a great FC and got looped into things, and now it seems to have stuck; the fact that I have people to do things with—or even just to chat with on the FC channel while doing my own thing—made me feel more like I had an in-game community or family, and that made all the difference. So I do absolutely understand where you're coming from.
If we were in the same datacenter I'd be happy to help in reaching out (and helping find other contacts to talk to), but unfortunately I'm on Primal and you're over on Crystal. Maybe some others in the thread here who are on servers in the Crystal data center can help reach out to you and help you make connections. I hope you have better luck this time around than you have previously.