Hm, keeping the story evolving around V and Jackie as mercs does seem possible and we could see a lot of drama, plot twists and some inner conflicts of the characters as well in such a story - it really depends on the talent of the storytellers, and REDs showed us already that they are good in this field. We would probably see all the cyberpunk tropes put into the scenario, maybe a huge focus on the conflict with the corpo world in the main plot; some tough choices for both of V and Jackie along the way; some sacrifices; meeting other people from Night City and badlands with their own dramas; alliances, betrayals; they could both learn during the story that the merc work and going towards the top is paved with much more obstacles and complexities than they initially thought, and that everything comes at a price.But also in story terms, I can't see how Act 1 could have been intended to carry on, instead of being deliberate setup for Act 2. Jackie is a likable but generic character, so is V, and if nothing is at stake except being rich and famous, well, that's not (in my opinion) really a game with a compelling narrative at all. It's more of a random adventures action game. I mean, where are those two characters supposed to go, storywise? They got rich? In story terms, that's surely just a huge shrug. Why would anyone care?
Not sure what the end of the whole tale could be... maybe one of them or some people close to them would have to perish to bring the necessary weight to the story; maybe even an evil choice for V to sacrifice his/her friend Jackie to get to the top, just to realise in the end that it was not worth it. Well, I guess many possibilities for a nice, complex and compelling story. All these philosophical questions that we have right now could be implemented to such story too (again, it depends on the talent of the storywriters).
BUT the story is good as it is right now, mainly because it's unique, with you trying to save your life, and having some opportunities to make the lives of others a bit better before you leave the world. It's also a good nudge for V to reflect on life, identity and values throughout the whole game from their own, personal perspective.
So yeah, I think it IS possible to have a good story without Johnny and rush for cure, but the tale we got is already very good, so no point in changing it in any way. Maybe REDs will provide us with some other approaches to stories in the next Cyberpunk game (if it ever comes).