It has been a consistent problems in games I play and this is the reason why. If you have a DMPC and that character is a core part of the party for balance/mechanics reasons there are really only 2 ways to play it.
1. One is the DMPC makes no decisions at all. In this case it is not really a character, it is just a set of skills the party uses to cover gaps. It rubber stamps whatever the characters do and the characters can use it however they want more or less. The party decides what magic items he gets, the party decides where they go, what they do. The character is essentially a party familiar or a meat shield.
2. The other type is the DMPC that makes core decisions and they can't really be vetoed because they need the DMPC to play the game. This is what you are talking about in the quote above. A realistic, fleshed out character is going to have opinions and there is going to be friction between that character and other characters, just like there is between PCs. As a DM though if you are playing this out there is a huge power imbalance.
There's a fairly big middle ground between your options 1 and 2 above; and yes DMPC decisions can (and sometimes should be!) vetoed, in that if the DM's doing it halfway right the DMPC will sometimes make a wrong decision or come up with a bad idea just like a PC would.
As both player and DM I'd never want them to be your option 1 above, and it won't be: the other party members see the NPC as every bit as much a party member as they are, and treat it as such. It gets a vote, it can claim treasure items just like any other character, and so on.
With option 2, just because an NPC belongs to the DM is no reason for it to have any greater voice in the party than anyone else.
As for there being occasional friction between the PCs and the NPCs, why not? There's also friendships, romances, rivalries, and all sorts of other interaction between PCs and in-party NPCs - why not friction and arguments as well.
An NPC that is part of the story is different. They are not there to round out the party, they are there for story purposes.
These are often the bigger headache as DM, in that you can't really just treat them like any other character and let 'em die if they're gonna die, as they're expected to survive until the key moment when they are needed for the story.
Also, if that story-based NPC is there for a nefarious reason it can be useful to have had other NPCs come and go as well such that this one doesn't seem out of place.