I know a lot of people don't want Death Note to get yet another adaptation, but I still think there's a good story to tell there.
It can easily be updated for today. How often do we see (everyday) people escape justice for crimes they clearly committed? Politicians, celebrities, police, the rich and powerful, and so on. You can make Light more sympathetic by making him kill people who are otherwise untouchable. People who probably deserve some sort of punishment, and wouldn't get it otherwise. But the question is whether or not that power belongs in the hands of any one individual - especially when (inevitably) that power corrupts.
In this instance, L would be a tool of the status quo. He's not wrong in that he's fighting a mass murderer who probably shouldn't be wielding such power, but on the other hand, he's also preventing any sort of meaningful change.
Of course, the change Light brings about would inevitably be based on fear, and that opens up that Watchmen-esque question about how can any peace based on the idea that society has a gun to its head be true peace (and that's even before Light's own ideals become twisted), but even so, L offers no other solution other than catch Light and return things to the way they are - a failing society where corruption is rewarded.
I think it could still work with only some slight updates. I think one great change the live-action Japanese TV drama made was Light hating the police. In that version, not only are the police were not only completely ineffective at stopping crime, but because his father is an officer, it meant that he wasn't around during the period when mom's Light got sick and died. Even after this, his dad still wasn't around due to his job, and he kind of ends up being the de facto parent to his younger sister. And he blames the police as an institution (and his father) for all of this, seeing them as ultimately useless.
Now, ideally a new remake would take this even further and would make Light aware of the corruption within the police as well (even if they make his dad "one of the good ones"), but even just keeping him apathetic towards them due to their inability to bring about real justice would help modernize it.
Light is never just a killer for the sake of killing, and he doesn't target randomly. At the start of the series when he first sees Ryuk, he believes he's there to take his soul, and he's fine with that trade. Even though he's only had the Death Note for a week, he feels that his own life is worth sacrificing if it can make a difference in the world. This new show would just need to play up that idealism, and show how it becomes twisted over time. Something which happens way too fast in the original. Maybe take a few queues from Breaking Bad, in that regard. Walter wasn't Heisenberg by episode 2, after all. Light shouldn't be full blown Kira that quickly either. Give that transformation some time to breathe, and be more subtle about it, so that by the end the audience feels that the Light from episode 1 and the Light from episode 100 are hardly the same person anymore. But even so, it happened so gradually that they barely even noticed the change.
If you really want to, you can take further inspiration from Breaking Bad and have Light come to terms with being Kira in the end, the way Walter does with being Heisenberg. Have him stop lying to himself and admit that in the end that he enjoyed the power trip. That on some level, playing Kira was always for himself, no matter how much he said he did it for the betterment of society. Even when he was willing to sacrifice his life to make a change, it was still for him because he was the one making that change.
And I know a lot of people love how pathetically he dies in the manga, and that's fine. But I don't want a carbon copy of the manga again. For all its mistakes, I appreciate that the Netflix Death Note was trying to forge its own identity and do something different. We've had the live action Japanese live action films, the live action TV drama, the musical stage play, and the anime, all of which (mostly) follow the manga with only minor alterations here and there. I want something truly different, that takes inspiration from the source material, and pays homage to it, but has something new to say, rather than simply recreating what's been done time and time again.
In any case, I'm always looking forward to more Death Note!