No Wandrian you're not wrong in thinking it's disconnected, because it truly feels that way.
The original DMC story is terrible, you can think it isn't all you want DarkRhythm, but it is. It can connect fine, i mean that's the basis for a good story, it has to flow together. But this barely does and it's bad. Because it was never meant to be good, because they didn't care.
God the DMC anime was awful too, friggin Madhouse ruining shit even though i KNOW they can do better. Love that soundtrack though.
As i'm running through this game as i type, by mission 10 i was more invested into the actual characters in this game than they even tried to make you in the first 3 games.
However i also miss old Dante...... sure new Dante is also witty and sarcastic and an ass. But DMC3 Dante was THE SHIT.
And the game was rebooted because the original series had run dry, the creator moved on so it's not like he was creating new ideas for his franchise. DMC4, although selling more than the original 3, still didn't net them a lot of money so Capcoms in house development team didn't want to waste time on it.
The only way another one was going to be made was if a 3rd party developer took the reigns and created something new. And so that's what we got.