This is going to be a bit of an echo but nevertheless

• The folks behind City of Mist (crunchy PbtA) are working on Tokyo: Otherscape, cyberpunk with mythic flavour. Haven't looked into it apart from following along the Kickstarter so cannot comment on how easy it is to remove the mythic elements from it. But yeah, in development so perhaps not what you need now.
• Cities without Numbers. Kevin Crawford's upcoming cyberpunk system which going to the printers soon, mid August, and then the pdfs will go on sale. Will come with, as usual, great GM tools. Since it uses the same engine as Stars/Worlds Without Numbers you can check out the bones of the system already now. And the basic editions are free.
• If you want something with a clear mission structure you cannot go wrong with something Forged in the Dark. Don't know what though but saw some in another reply.
• Shadow of the Beanstalk for Genesys. Genesys is great featuring awesome dice, but those can be a but hard to get. Haven't read SotB but will echo the recommendation.
• Among the "pure" PbtAs there is at least the Sprawl (classic anti-corp cyberpunk) and the Veil which is a bit more artistic and philosophical (my impression).
• (Long shot) While Ironsworn: Starforged is very sci-fi it has a great mission structure (vows), ways to track progress (best trackers ever!) and tools for emergent gameplay (gm-less play). With a bit of reflavour and hacking in hacking it could be great. Bet there already is a cyberpunk hack for it...

we have dropped initiative, the XP system, and the very, very detailed way it handles "damage to body parts."

Yeah, I think it might be a good idea to at least look at alternatives.

We play Shadowrun, so probably there are better pure Cyberpunk systems. But I think it should be enough to just cross out the magic parts to get what you need. Some of these might be available on itch.io only

• The Sprawl (PbtA)
• Karma in the Dark/Runners in the Shadows/Subway Runners (FitD)
• Shadow of the Beanstalk (Genesys)
• Neon City Overdrive (City of Mist)
• Interface Zero (Fate/SWADE)
• Sprawlrunners (SWADE)
• there are multiple non-official Shadowrun hacks for SWADE (check forums etc)
• Megacity Mercenaries
• vs. MIRRORSHADES
• Rainworld
• Axon Punk Overdrive
• Bookmark no HP Cyberpunk
• CyberPunk One Page (lasers & feelings)
• Running in New Orleans (Dungeon World)

I love cyberpunk and have run many different systems and settings over the years. A few years ago I picked up The Sprawl a PBtA book and it clicked. Of course very narrative but with the cyberpunk stuff you want- hacking, gear, cyberware but without a ton of book keeping.

Is it perfect? No, but it's a good fit if you like to narrative games. It's default is mission based gameplay, but have done several campains breaking out of that mission structure. Doing a Shadowrun campain right now :)

I can't recommend it enought! DM me happy to chat more!

http://www.ardens.org/games/the-sprawl/

Even if you aren't interested in OSR style gameplay (which sounds like you ain't), I'm still gonna recommend Cities Without Number because it's sure to be a dearth of fantastic tools for generating campaigns in a cyberpunk environment.

I'm not familiar enough with the Forged in the Dark stable of games, but given that Blades in the Dark is about thieves and their schemes, I would be extremely surprised if there isn't a cyberpunk version, which seems like it would be right down your alley!

There's an adaptation for Fate for Eclipse Phase. Mindjammer is also unsing Fate.

If forged in the dark is close enough for you of PbtA, I am reading Scum and Villainy but it could be more Space Op than Cyberpunk.

I've looked for any PbtA games, but there seem to be none in the cyberpunk genre.

The Sprawl is Pondsmith cyberpunk in PbtA. It seems like it's close to what you're playing. The Veil is a japanese style cyberpunk, closer to Ghost in the Shell.