In fairness, it's difficult for the story to account for the possible skill level of the player, of course, so if one dies tons of times or even just barely holds on during said action beats, the exasperation Lara expresses can be a deal more accurate. It also comes full circle about halfway through, when Lara starts losing herself in the bloodshed the island forces her to resort to, eventually to the point of Badass Boasts that she's going to hunt the Mooks down and kill them all. It still applies early on, of course, since not five minutes after Lara agonizes over her first kill, she's gunning down dozens of scavengers by sheer necessity, though it does gel with the narrative in a sense that the island is forcing her to adapt or die, so if she did let herself dwell on it for too long the game would end a lot sooner.
It's less a matter of player skill level than it is the gameplay that's presented to the player that's dissonant with the story. This is certainly not as much of an issue after Lara decides to actively fight back upon reaching the Shanty Town sequence—because the narrative indicates she's accepting the death and killing as a necessary part of her environment—but the moment the player gets a gun in their hands the game begins throwing dozens of mooks at Lara at a time. The player is given no alternative but slug it out, which is at odds with Lara's indicated mindset in the first half of the story, with only her Handwave of the situation as an explanation.
It can be argued that the very nature of the first kill actually helped her adapt quickly to having to kill since it was so brutally up close and intimate plus clearly showing her just how dangerous the island's inhabitants are. Compared to such an intense moment, shooting at later enemies from cover could just come that much easier for her (as she later states). Certain unlockable abilities do support her growing as a killer such as being able stun enemies with dirt generally being unlocked before being able to injure or kill in the same circumstances, being only able to shove nearby enemies for a few hours before gaining a melee attack, and the stealth takedown even changes from a choke-out with the bow to driving the climbing axe into the necks of mooks. Compared to others games, the ability progression actually does a pretty good job of supporting the storyline.
Another example with Lara's equipment upgrades: While some of the modifications (such as duct taping the machine gun clips together to make reloading faster) are practical and completely plausible with what Lara has on hand, others defy Willing Suspension of Disbelief. Like Lara somehow being able to convert a Type-100 submachine gun into an ''StG-44'' assault rifle. Some of these upgrades would require not only specialized equipment and knowledge to turn out, but Lara is able to do this using random pieces of scrap metal and animal parts (remember, you get salvage from the animals you kill) at makeshift camps with no formal training. Rule of Fun can be used to Handwave this, but it doesn't stop things from getting to be a fairly egregious example of this trope. (Of course, given Lara's upgraded pistol was given to her by Roth, it's possible she instead found an intact StG-44 in one of the crates and swapped to it. Still won't explain how the Type-100 and StG-44 can use the same ammo.)