I think there is a way around no-regeneration-without-cyberware issue. Make it so that each level has its own advantages, the lower difficulty level you take, the more fancy regenerating tech your character owns from the start while those who take "Cyberpunk" level won't get any and will need to actually buy one for themselves. Since game is supposed to be heavily focused on single-player everyone could be able to play what one desires.
You could use bandages if you're only slightly damaged and such damage would heal up, slowly, over time. Permanent damage would require more serious work, like buying yourself a new arm, some professional doc, etc. Healing capabilities of a human body does indeed has limits. Armor would basically help you keeping yourself at low level of damage from where you can regenerate it. How long should soft-damage regeneration take? Again, I'd say it should depend on difficulty (lower means faster regeneration, higher slower, depending on how hardcore one wishes to be). Either that or the game has to force part of potential playerbase into something people wouldn't find good enough (either too easy or too hard).