So months ago I created a list for myself to see if I was being stupid and whether or not Luke in TLJ works. Here it goes!
To understand Luke in TLJ, I decided to start with what TFA tells us about him.
1) Luke has disappeared! Literally the first sentence in the title crawl. Very obvious and self explanatory.

(the wording suggests that his disappearance is recent, and factoring in Luke's and Ben's appearances in TLJ, 1 to 3 years I'd guess)
2) Luke was training a new generation of Jedi.
3) A student, Ben Solo, turned on him destroying everything he had worked towards.
4) He felt responsible, and walked away from it all.
5) No one knows where exactly Luke went, which means he didn't tell anyone. This tells us that he's very serious about his exile. And doesn't want to be found. And that his exile is very different from Obi-Wan's or Yoda's.( Bail Organa knew where both Jedi masters were)
Now that's out of the way, I could figure out how Luke should be. I did this by asking three questions.
1) why has Luke gone to that island?
2) why has stayed?
3) will he come back, and when?
So the first question. Why has Luke abandoned his friends and loved ones, when they need him the most? AND not tell anyone where he's gone!? This is Luke Skywalker we are talking about. He will recklessly charge in head first to save his friends. What could possibly change that? For me there's only one answer.
Something very very terrible happened. I mean it has to be bad. And that's what we get. Luke students are murdered, his temple burnt to the ground, some of his students have fallen to the Dark Side. And the cherry on top, his nephew is the one that did it all. And he, Luke Skywalker, Jedi Kinight, was catalyst, the guy who knocked down the first domino. He accelerated Ben's fall. And in my mind that is sufficient enough for Luke to leave for this island like we see.
Next question, why has he stayed for 1 to 3 years on this island? Again, when his friends and loved ones need him most, and he didn't tell anyone. Here's a some of options.
A) He's gone there to find some new knowledge.
B) He's being tempted by the Dark Side so much, that if he were to return to the fight he would fall. (this one isn't my idea, somebody on the RPF gave it to me! Thanks!)
C) Something is physically and/or spiritually holding him there.(also not my idea, thanks again RPF person who I can't remember, sorry!)
D) Is having a bit of Jedi walkabout, perhaps tied in with option (A)
E) He truly believes he's helping the galaxy by not being involved. He believes he's helping his friends and loved ones more by staying out of the fight. This comes with the idea that Jedi Order aren't doing a stellar job.
Next I broke down those questions. Separating what I thought worked and what didn't.
A) So I actually think this might have been apart of his motivation originally. But to have this be his only motivation, makes Luke seem very selfish. Just think about Luke saying "I need to find out what went wrong. So I'm leaving and not telling anyone. Not even my sister and I'm not going to leave a message with R2, that way I can study without people bothering me for 3 years." Yeah that doesn't work out for me. If he had told someone what he was up to. Like a simple message sent by R2 like "Leia, I've failed you, I've failed Han, I've failed Ben I'm sorry. I'm leaving now to see if I can find out what went wrong." If there was message like that, then this would work for me. But we don't get anything like this.
B) I really really like this one. But considering how people think he shouldn't be tempted by the Dark Side ever again.......this wouldn't work for a lot people.
C) I also like this idea. But again why didn't he tell anyone? And why is he being trapped at an old Jedi temple? If he'd gone to a Sith temple or something, this option would work.
D) Like option (A) this makes Luke seem incredibly selfish. Just imagine him saying "I needed some ME time, to clear my head......." Heck no! Maybe something like this could have worked, but we still would have needed some sort of communication from Luke.
E) This option is of course the one we got in the film. I feel it's the best, because it gives Luke a very good reason to stay. If he believes he's doing more good out of the fight, then in the fight. It makes his exile seem less self serving, and more selfless.
Third question. Will Luke return and when?
Only really three options here.
1) Comes back to rejoin the fight immediately.
2) Resists at first, but eventually comes back.
3) Never returns, stays on that island.
And the break down.
1) This result only works for option (C) seeing as how he'd probably be super ready to return after being freed. And it kinda has to be used for options (A) and (D). Because there isn't a good reason for him to stay. But at the same time it induces a little problem. That problem being, it makes Luke into a diva like, very fickle person. The idea that Luke would disappear for a couple of years without telling anyone to have some "me" time. And then when Rey shows up, to have him go "Okay so I ran off without telling anyone, for a couple of years. Han got killed by his son. But that's the past, let's not dwell on that. I've had my 'me' time, so let's go save the galaxy!" Nope, doesn't work for me. And for this reason this definitely can't be used for option (B) or (E)
2) This option is what we have in the movie. Its the safest way to do things. Without making Luke shallower then a kiddie wading pool.
3) For obvious reasons this one would have never worked. Though a combination of (2) and (3) is what I would have liked. I would have Luke to remain exiled and removed. And he wouldn't comeback until the ep9. This would have really made people angry, but the payoff in the next film would have been worth it!