I personally sometimes do like simple action focused stories myself, and I don't think you need evil for it. Saying evil doesn't exist doesn't mean you have to think about every opponent you come across.
That's an axiom we don't share. And that's nice because your formulated the disconnect early in your demonstration and in a way that makes it quite clear that we're approaching the situation from different starting points.
For me, if you're roleplaying a character, you need to act according the moral compass you're roleplaying. If you're roleplaying a scum, you can probably kill people for no reason (or because it suits you, like you need gold and he has a purse). If you're roleplaying a character with an alien morality, you should endeavour to adhere to it. But if you're roleplaying a character who is "conventionnally good by modern standards[*]", then you need to think about every enemy you come across. Are you acting in self-defence? Is your action proportionate? Is there any other way to deal with the problem outside of killing him? Honestly, if your main way to deal with opponents is lethal unprovoked violence, I'd say that you're roleplaying an evil character.
[*]
Assuming there is a modern shared morality...
A good guy would think about every person he meets, like we do in real life, and using lethal force as the only last resort. Because good doesn't kill. Good helps the misguided to redeem themselves, after potentially dealing an appropriate punishment decided outside of the spot of the moment, which takes time and reflection, and by people not involved with the initial action.
So yes, not having something classified as evil requires you to think about every opponent, unless you're classified as evil yourself.
Being classified as evil allows indiscriminate violence. Just like in real life. We deem smallpox to be "KoS". We determined that its continued existence is antithetical to ours, and we eradicated it. It certainly wasn't evil to do so, because we were justified in that, by the fact that smallpox had no way to change its behaviour. In a fantasy game, determining that human woodcutters need to be dealt with because they are cutting down the trees that store the souls of your loved ones until they move on to the afterlife can be dealt by educating them and stopping them. You don't need to kill all humans, because they wouldn't be essentially evil. They would be doing an evil act, woodcutting your loved ones, but they wouldn't be essentially evil. So, when you meet a woodcutter, you'd need to think a lot about it. Is he doing it knowingly, or not? Is he doing that because his legitimate authority forced him to do so? And so on. You can't just see someone cutting down a birch tree and roll initiative, despite the gravity of the crime.
If he is under a curse to cut wood because a God created him in a way that he can't help but cut birch trees, then well, if there is no way to lift the curse and he can't help cutting birch tree because the curse override his free will, then activating your McGuffin to remove all of them would be more justified.
Just because the swordsman in the Indiana Jones movie isn't evil by nature doesn't mean you have to wonder about his motivation and discuss his moral standing.
The swordman is actually threatening Indiana Jones and is telegraphed as dangerously skilled. Of course Indiana Jones had to wonder about his moral standing. It took a split second: he was assaulting him and therefore Indiana Jones was justified to act in self-defence. If the swordman was attacking him because Indiana Jones had stolen an idol from his temple a while ago, Indiana Jones would be a scum for adding killing to theft. And to lessen his scum aspect for the theft of cultural artifact that we, as 2025 viewers might see as problematic despite being representative of what a 1930 archeologist would do, his theft is "justified" by the idea that he's saving archeological goods from traffickers or Nazis.
Most human beings enjoy violence to some extent, as long as no one actually gets hurt in the real world. This is a natural part of human nature that can be explored responsibly. It can be escapism pure and simple. This naturally begs the question I'm sure some of you are thinking, if you're not thinking of the implications all the time, what is the difference between an opponent being evil or not?
Well, this questions is solved by your initial axiom: "having no evil doesn't mean you have to take into account the situation of every opponent". So you initially removed the main difference between having evil or not.
In this case, you probably don't need evil in your campaign. But not everyone adhere to your axiom.
Let's take a simple example of the stereotypical evil orc game. Zoomed in, viewed encounter by encounter it may look no different weather they're evil or not.
If we accept your idea that good character can kill people wantonly because it suits them. If you see, say, clichéed evil orcs capturing the daughter of your village chief to sacrifice her to their god, and there is no evil, when you infiltrate the orc fortress and are seen by orc janitors who are about to sound the alarm, and there is absolute evil and the orcs are it, you can kill the janitors. If there is not, there is a strong chance the janitor is just a janitor. He has nothing to do with the human sacrifice planned by the orc leader. He's just an employee working for the wrong guys. He might not even know they are into human sacrifices. If you're invading their office and kill them, you're probably not better than the human sacrificers in the first place. You'll probably be able to rationalize your evil act by saying you had no choice, and after all you only followed orders from your village chief's and that you served the greater good by killing 3 janitors to save a single chief's daughter somehow. But without essential evil to cover your act, there is no way your janitorial assault wasn't evil in the first place. You killed 3 janitors because it was
convenient. You're roleplaying
murderers.
(Which is fine. It can be cathartic to roleplay people with vastly different moral norms as ours, the same it is fun to read about Greek heroes or Roland or... but we don't pretend that they are nice guys.)
Then near campaign's end your party gets a mcguffin that allows you to take care of the orc problem, permanently. You do it... right?
No. If you find an orc-genociding weapon and use it while knowing that orcs are people like you and me, you're evil. Unless there is evil, and orcs are established as smallpox, in which case of course you can do that, as humanity did with smallpox, without qualm. And that doesn't make you evil.
By the logic of the campaign, it's the only reasonable conclusion. Orcs are pure evil, they can only hurt everyone in the world and never be redeemed. It is the right decision, you couldn't even argue otherwise. No matter how correct this line of thinking is in the game world, for the people playing the game this can be a disturbing and dark conclusion. This is the disconnect, having true, pure evil species in a game puts you on a dark path.
No, it puts you on the path where you can eradicate them without asking too many questions every single time.
I think most people can agree it's fine to kill a fly in the real world, but it's a lot less fine if someone traps and tortures the fly. Plucking it's wings off under a magifying glass to prolong it's suffering. Laughing at it and encouraging people around them to watch it twich. Likewise it's fine to have consequence free violence in a game, less fine if someone is enjoying the pain and suffering they are doing in game, and something being evil is the best excuse for this behavior. I'm sure most DM's have a line with this sort of thing and will not let it go too far, but again, having evil in the game puts you on this dark path. Even if you don't follow it yourself, it is there and it encourages you to walk further down it.
Hum, no? In real life, there are people that think that killing people is right under some circumstances (they call that the death penalty) and I am pretty sure not all of them are into torture because of that? Having in game a situation where something is irredeemably evil (because of mystical circumstances, since we have no comparison in real life) like a curse (vampire, werewolves, elves...) or being created for that (demons, orcs...) make you sure that they can be killed because there is no possible redemption. That's the same reasoning as applying the death penalty in real life, except that you can be certain that there is no possible redemption in the future, and that doesn't seem to turn people from country who practice it into torture-hungry sociopaths. I am not sure the "if there is cause to justify killing, you'll end up enjoying pain and suffering" argument is grounded in reality.