The problem with a tight, linear story like we have is that it makes 0 sense in the context of the rest of the world. Think about it, you're given a literal deadline of few weeks, why would you go mess around with random gigs or personal relationships? If YOU had a few weeks to live, would you try to prevent that fate OR would you go diving with your girlfriend?
That's a classic videogame issue with stories trying to give you a sense of urgency. Same with the Witcher 3, you were on the brink of a war you didn't see happening, you had this urgency about finding Ciri but could play Gwent for hours instead...
The open world is here just for you to take breaks from the main story, but it isn't here for you to live in, at least this isn't how CDPR have made their two open worlds (W3 & CP2077) until now.
Story comes first, open world is just here to provide for a beautiful background painting, expand a bit on story elements, and give you a better taste of what it means to be living in such a world by encountering some interesting side characters. It provides you with a bit more freedom in how you want to tackle the story (choosing the order of your quests, where to go first, etc).
But at the very core, for CDPR, and you see it in the Witcher 3 and CP2077, the story comes first and the world comes second. They never intended to make it a sandboxy thing where you play however you want with tons of dynamic stuff and have a side story you may or may not follow. This is a narrative driven experience first and foremost, they didn't lie about this and this is kind of their trademark. It's the same with The Witcher 3 (everyone's entitled to their own opinion though to say if it's a good or bad story driven experience).
I do feel though, and I might join you on that, that the random gigs and such side activities would have been better integrated if the game did a better job in opening slowly the city to us. Like this 6 months ellispe people complain about with Jackie ? I have the feeling this was meant to be the tutorial and to make you do pretty much all the gigs and NCPD calls in Watson.
Why did they cut it ? Maybe because some would have complained about being forced to do side jobs for a long time before the story starts kicking in (imagine having a quest like "complete 5 gigs before talking to Jackie" or "earn more streetcred before going to Misty" - no one would have liked that). Maybe it was something else entirely, I don't know.
Personally, I did clean all of Watson before taking on the Heist mission so, I was feeling pretty "connected" to the story. I guess that helped tremendously in feeling my V was already someone well established and kind of known around the city by the moment the title screen appeared.
But I can't help thinking that a "tutorial/prologue area" (such as White Orchard - it would have been Watson here) is lacking in CP2077. It would have helped managing the story pacing, which can feel really overwhelming and rushed if you focus on it.
And as a side note, I would have loved a better way of handing to you those gigs, rather than a point on your map and a phone call. The game lacks a sense of exploration. It would feel way more organic if you just walked through the streets and have one of those gigs/NCPD stuff randomly opening to you because you happened to be on that street at that particular hour of the day. Right now, it feels static.