Fighters are pretty much the bread and butter when it comes towards combat. That's no exception in lotfp.
Fighters have the greatest hit die of the human classes and are allowed to start play with a minimum of 8 hit points.
Additionally they start with a +2 Base Attack Bonus and access to the Press, Defensive, and Parry +4 Combat Options.
Besides gaining more hit points and better saves when he gains a level, the fighter's base attack bonus increases by +1 per level. This makes the fighter deal more consistent damage towards the end of his adventures without turning him into an unstoppable juggernaut.
The role of fighters in 5e is treated much the same as Lotfp.
Fighting Style allows them to customize their mode of attack.
Second Wind allows them to press on after taking a walloping.
Action Surge allows the fighter to unleash bursts of damage, especially after he's gained extra attacks, or accomplish more complex tasks in a single turn.
Out of all the classes in 5e, fighters get the most Ability Score Improvements and Extra Attacks.
Towards the end of their adventure they earn Indomitable, which allows them to succeed on potentially world-shattering saves.
Not to mention the Martial Archetypes allow the fighter to further his tactics.
Champions appear to be raw stat engines with their increased crit range and ability to regenerate later on.
Battle Masters interact with their allies and enemies via maneuvers and Know Your Enemy can lead to some interesting situations.
Last but not least the Eldritch Knight allows the fighter to dabble into magic, giving them a summonable weapon and the ability to more effectively blend combat and magic.
I've heard the particular leveling process utilized by Lotfp and 5e referred to as Scale and Gate respectively.
In Lotfp when a character levels up, he gets more hit points, better saving throws, and an increase towards a tertiary stat. This would be base attack bonus for fighter, spells for magic-users/clerics, and skill points for specialists. In other words the character scales better as he levels up.
When a character levels up in 5e he gains access to new features that add a new paradigm into the system, or he gets upgrades to already existing features. The leveling process is the gate that prevents lower level characters from being on par with higher level ones. The characters still receive stat upgrades but it is distributed sparsely throughout the character's progression.
A level 1 fighter in lotfp is not that different from a level 13 fighter if you completely ignore stats. But a 5e fighter at level 1 looks completely alien from a level 20 fighter by their features alone.
I don't believe that one particular style is better than the other and a "scale" vs "gate" argument isn't necessary.
I think both systems do a good job with their own style and I find it difficult to gleam any hidden lesson by comparing the two since they're so different from each other to begin with.