I've complained quite a bit about how Tomb Raider 2013 starts with some interesting ideas but ends up foundering on the crass and rote machinery of contemporary AAA games. I am not sure how you would "fix" the game, given as many of the potential ideas it introduces are of a sort that no-one has yet figured out how to turn into a proper game mechanic. And if you removed the cover-shooter combat fest, the remaining game would be too short.
You could, however, re-tell the story in a form that does allow better use of the archaeological problem-solving, and survivalism that goes beyond simply hitting a few quick-time events (let's put it this way...starting a fire to survive her first night is given in a cutscene. Period. There isn't even a QTE to react to. Your involvement as player in this basic act of survival is quite nill.)
So given that, what would change?
Well, for once thing, stop with the constant combat. It should have been possible for the game to offer stealth as a way of avoiding combat entirely, not merely winning it (and for that matter, all of the stealth combat opportunities are basically just as easy to win with a straight-up fight.) In any case, in our fic we can give her other options.
Cut back on the horror of the Solarii cult. Yeah, this is wonderful visceral stuff but if you've set up this internal plot backbone of Sam being threatened first with immolation, then possession by a body-surfing ancient queen, and you've got Lara doing her only serious in-game archaeological/anthropological work figuring out the truth of Queen Himiko, then you do no favors to nobody by having the Solarii already upping the ante in mass torture and complex ritualistic behaviors and massive construction projects and a long and complex history. It just distracts. No, more; it makes Lara's worries about Himiko and her interest in Sam seem like missing the point.
I mean, really. It takes Lara half the game to realize the cult might -- gasp! -- put Sam to a fire ritual. So...she didn't worry for Sam's sake before that? Like, she sort of forgot that she'd found the strung-up body of Stephanie not two hours after landing on the island?
And then there's all this legends about the Oni, who might be skulking out there and might be dangerous. With their bows and old swords. Somewhere up in the mountains...past the hundreds of men in full body armor with pintle-mounted machine guns who have already systematically slaughtered dozens of her fellow crew members.
And, yeah, so really. So the big Oni eats people. That's both scary and disgusting, and they've got that undead thing going for them, too. But seriously -- the Solarii are out there torturing every shipwreck survivor to death, often burning them alive (and what they do with the ones they don't burn....ew.) Supernatural horrors just don't measure up to the horrors the much nearer enemy has been inflicting, especially when the Solarii are killing her friends in front of her eyes.
Seriously, if you removed the Solarii from the game completely it would be a stronger plot. Lara has plenty on her plate trying to survive the wilderness and the wolves and her injuries from the wreck and the magical storms Himiko is sending, and there is plenty of threat in the Oni (who in-game took out a W.W.II Japanese garrison with little effort) and the threat to Sam is enough to properly motivate her.
Of course you don't have to go that far. Mathias is interesting enough and dangerous all by his lonesome. And you could even give him a few buddies. You just don't need to give him a tiny island civilization of hundreds of men, and you certainly don't need to have those hundreds of men all out pacing the trails hoping to take a shot at Lara.
With more exploration and more archaeology, the tombs matter. And if we aren't maneuvering Lara into set-piece battle after set-piece battle, plus positioning her remaining friends in the best places to play Ten Little Indians on them, then we don't need to go running around back and forth quite so often. Which means fewer convenient knocked-on-the-head-in-cutscene moments.
And also means she doesn't have to run into Roth and Grim and Sam a half-dozen times only to lose them again. She can be serious about the solo survival thing. And, yes...the Kenobi role is so firmly genre-established, Roth is still likely to die, but it isn't quite as necessary. And without the grim-at-all-costs and always-on combat vibe the actual game established, we can afford to leave other survivors alive.
Or not have them in the first place. The various other characters are really a bit of a distraction. For all the actual interaction she has, they might as well stay on the other end of a radio, and the other side of the island as Lara struggles to get to them (or to find answers).
Aside from Whitman, that is. Because if we aren't making this big thing about Solarii leaping out and going "boo" every ten minutes, and killing someone in front of her every now and then to keep her focused, then we can spend our time really working with the difficult relationship she has with Whitman. This can be as isolating, as much a pressure on her and a challenge she has to overcome, as any bearded man with a bow full of flaming arrows. He is her supervisor, the civilian head of the expedition, her academic advisor, someone with an academic reputation (and who could destroy hers). That is a lot of good meat, at least to me.
And he's interesting enough (if a stock character) to take a little more time himself. He's too aware of not being where he wants to be, of pushing harder and harder chasing down even more unlikely rumors in hopes of making a comeback. And facing domestic crisis of the sort that shouts "midlife crisis" as well. Making a young, female, attractive, and extremely skilled and dangerously up-and-coming student a foil he can hardly avoid clashing with.
And add to this mix that Himiko is both quite real and quite mad and creates a situation which is far from merely academic. And add to that Matthias, who is grounded enough to also realize that Himiko's powers are real but who can play the open-but-skeptical academic well enough to sucker Whitman along...
And, hell. This also leaves our potential pool of survivors -- Jonah and Reyes and even Alex work well for this -- tinkering with the boat almost through the story. Spending a good chunk of it seeing nothing more hostile than a cold, windswept beach and the effort to salvage what they need from the copious wrecks, and at first almost hostile to Lara's claims that there is more on the island than they think. At least, until Whitman leaves with Sam...
Can even have Roth skulking his way down from the backside of the island himself, performing his own recce, a voice on the radio for most of the story, helping Lara only with emotional support.
So this builds the solo survivor story in essentially three phases;
Phase 1 is Lara alone. She gets a brief hope spot when Roth manages to contact her on the radio, but basically she is on her own trying to survive. Whether the first-night encounters with Sam, Matthias, and some wolves occur is pretty much up to how well you can combine that with the original "Scavenger's Cave" scenario and make it a part of the confusion within the first few hours after the wreck.
Phase 2 she hooks up with Whitman, perhaps others, and what she has to fight now is the academic and professional barriers; the natural subservience to an older, established, male authority and the academic bastions and official histories he represents. So here, she has to learn to trust herself and her own instincts even when there is the option of merely tagging along.
Phase 3, she's learned enough about Mathias and Himiko to be wary, and concerned for her friends, and this is the true rise of the Tomb Raider in that she choses to leave the comfort of the camp on the beach and the certainty of the others there to find the real answers.
And I sure wish someone would write THAT story. Because I'm not going to!
(Oddly enough, Queen Himiko got a mention in the novel I wrote almost two decades ago. Historical domain character, after all. Even if the world of Shirato is an analog, not really the Japan of our world.)