I am as of late on a Reborn marathon, having completed from the Ice Gym up through Titania in like a week, so my theory gears are all wound up. Sadly, as you say, there isn't much evidence to what is up with Lin, but there's plenty of evidence that SOMETHING is off. Things that immediately come to mind and are big red flags are Anna not seeing Lin, her immortality, the age paradox (which may just be my confusion, I'll address that later), and how she seemingly works AROUND the player rather than against in a way that seems to go beyond protag-armor (such as stating it is important that the player leaves with the bracelets). If interpreted correctly, from these points, we know that Lin is at least biologically and physically immortal, has some inhuman trait about her (whether it being demonically soul-less or just not actually human), and we have something to do with her...
I will also be 100% honest; as someone who loves predicting the future or explaining present traits of a game or the characters in it through theorization and imagery...Lin and the protagonist have me stumped...my best guess for Lin at this point is that the current Lin is some sort of physical embodiment of a force or magic. If I understand the story correctly, there is one VERY weird time paradox: how could Lin be a child younger than Anna, go on to be adopted by Terra's family for her to corrupt Terra, and now be an adult all while Anna and Noel are still children? What I think is that after she electrocuted Dr. Connel and he got rid of her, Lin as a child somehow made her way to the Arceus chamber under the grand stairway (or to some other holy/un-holy place). From there, I think some godly force or evil corrupted her body and is currently using her as a vessel. This makes sense because it would explain her seemingly ageless and immortal traits, could explain the weird voice in that area as the true child Lin that was exiled from the body that was once hers and now is used by this "evil", could be why Anna doesn't see anybody (as there is no soul in "Lin" now, only some evil force), AND in many forms of media, possessed people often have issues functioning properly (such as narcolepsy).
As for the protagonist, I feel like we are going to get some sort of Undertale-grade reveal (as in very metaphoric and supernatural). I feel like that we are something INCREDIBLY integral to the Reborn universe beyond just being the protagonist. The biggest things that suggest this to me are the unexplained and coincidental nature of our arrival (random inexperienced trainer brought to a dying region (oh also, I think in-game Ame knew this too) at the last second who is suddenly the largest hero)), Anna's discussion of our aura, and (based on if Lin is some sort of fucked up, supernatural creature) how Lin treats us to the confusion of other Team Meteor members and other supporting characters (such as seemingly letting us go in Tanzan or pretending we are a team and letting us keep the bracelets in Titania's Gym). This, mixed with the odd, reality-changing nature of the couple prophecies in the current game, lead me to believe the dynamic between us and Lin is going to be something world-shattering.
Tl;Dr: The mysterious nature and backstory of Lin mixed with the themes of rebirth and supernatural power lead me to believe that Lin's body is home to some name-less evil force that exiled young Lin from her body (in order to explain the age paradox). If this is at least somewhat true, I believe that the foil that is us will also be revealed to be something of similar power/supernatural ability.