The problem is that none of these new streets will function as actual streets. For the most part they don't connect to anything else so there isn't going to be any through traffic. You're better off calming the arteries which ring the site and creating more obvious and attractive entrances into the central court.
That is if I thought the building should be demolished, which I don't. Anything that replaces it won't have an iota of ambition that Rudolph had. Far too many people here value height over anything else and the way this city has developed over the last 20 years proves that actual architecture has been discarded. The GSC is faaarrrr from perfect but it shouldn't be torn down just to be replaced by some bland mid rise that no one will think twice about.
I don't see them being arteries for through car traffic. More as a means to create a more human scale in the area and break up the superblock for pedestrians while also not completely ignoring cars. I'm sure that anyone with some semblance of a traffic engineering background would call that MS Paint mock-up absurd.
That said, I did consider that you would get some use out of the Cardinal O'Connell extension. It would enable people entering/exiting MGH to avoid the Merrimac/Staniford mess. I also see it as a good pedestrian connection to much of MGH from Bowdoin. Currently, it's much more circuitous to reach anything other than the buildings directly along Cambridge St.
The Bullfinch Place connection is really to further break the block up and provide a direct Pedestrian corridor rather than a through street. Apart from cop cars and mail vans, it will be quiet. But there's nothing wrong with a quiet street.
I also am not fully on board team "knock it all down!" I actually would stick my neck out to fight to preserve/fix the Lindemann side of things. I'm 50/50 on the Hurley, my biggest issue being the sheer mass of the thing and how anti-urban it is. Not so much an issue with design. It's a shame Brutalism came about in an era of auto-centric planning because I really like it it from a design standpoint. If there was a way to maybe open up some pedestrian walkways between Staniford and New Chardon, add some retail/street-level activation, etc. AND save the building, I'd be all for preserving it. Especially if you could open up the plaza and the New Chardon side for development.