As you point out, it's more that he doesn't really seem to be progressing in a significant way in any area.
To really thrive in his current role he would need to become more of an instigator / playmaker on defense like Smart, or he would need to add a lot of ballhandling and playmaking to his skillset to fill a role more like what Hayward is doing.
Or I suppose he could go the way of Morris and really work on his 1-on-1 scoring.
Problem is, Jaylen still isn't very good at finishing inside, creating space for his own shot, navigating traffic with the ball, or finding open guys.
So I keep seeing him as maybe a taller, more athletic Jae Crowder. But to become that he would need to improve big time in the one area that really set Crowder apart --- super quick decision making when he gets the ball.
Crowder would catch a pass on the perimeter and then either shoot or immediately burst to the hole.
Jaylen never makes quick decisions on the floor, he always looks like he's thinking in real time.
Sometimes being cerebral is a bad thing. Point guards can be cerebral because they can dribble around and probe the defense, looking for openings.
Wing players with poor handles can't afford to be cerebral.
The thing is part of whether you think he has improved depends on how you view the start to the season. As the OP pointed out he is actually shooting slightly better in many area of the floor, including at the rim in which he was at 63% (before last night, may have changed). He was better form 3-10 and 10-16 as well. His per 36 numbers are all pretty much the same as last year. But given how badly he started out that's actually fairly impressive. He was shooting like 47% at the rime through the first month, 25% form three, and wasn't rebounding either. The last two months those numbers have all risen back not to last years number, but last playoffs numbers on a per 36 basis. And those were excellent numbers. The numbers are just low because he started out horribly. So if the last two months Jaylen is closer to what he is than the first months Jaylen, and I think that's not impossible due to the injuries he had and adjusting to role, then he has improved as leats in some area.
Now first of all, I believe he is better at creating shots for himself than most here believe. He is probably the best driver on our team, and that is a type of (generally good) shot. He has always had a decent post game and he now has a nice fade away he uses when he gets into the lane and can't get to the rim. I think even his decision making has improved, he sees a lot more kick outs then last year, and has learned to dump it off to the big inside. He certainly loses his dribble A LOT LESS on the drive than he did as a rookie or even early last year.
The main problem is he just doesn't get the opportunities for proper evaluation on this team. I can't know how good a ball handler he is really is because he never gets to run a pick and roll, like at all . If you told me he got more than 1 a game I would be shocked. I can't know how good a playmaker he is because most of the time when he touches the ball its a one on one situation where he ran out of the corner to get the ball, everybody else on the team is stagnant, and Jaylen tries to beat his man or give it back to the point guard. He has long 4-5 stretches where he sits in the corner. And I don't blame Brad for this really, he has better options (or at least safer, coaches always prefer to go with the guys they KNOW can do something, rather than guys who could. Same reason Tatum doesn't run PnR even tho he has some talent for it). But the general idea is to know what he is we have to see him get some opportunity, and for him to improve as a passer he has to get some game reps at it and be allowed to make a mistakes. Maybe that never happens in Boston I don't know, but its hard for me to say he can't do something when he ever gets a real opportunity. But just form flashes, I think some things have improved.
In short, put Jaylen in the PnR or PnP and lets see what he can do.