Not their authority. The rules. Unless you're saying the rules = DM authority, in which case d20 + modifiers = or greater than AC to hit is also the DM using his authority.
No, the PHB and the DMG clearly give the GM the authority to decide, but do not provide any operationalization of this. There's no rule you can point to and say, "here is where a paladin has a bad thing happen to them for not following their oath." Nor can you find where a paladin has crossed a line to become an Oathbreaker, just that such things exist somewhere. So, what you have here is an assignment of authority -- the GM decides what becomes of a paladin and their oaths -- and a loose constraint -- the GM may decide bad things if you don't follow your Oath. I say loose here because there's very little to say what following your Oath means in any given situations -- it's rather subjective. What did happen here is that the authority to decide what happens with character build, usually a player authority, has be explicitly reassigned to the GM in this case. That's the extent of the rules -- GM says.
Largely, a lot of 5e can be summed up this way. Not a bad thing.
The DM is not adding constraints. Those constraints are put into place by the rules.. They're called Breaking Your Oath and Oathbreaker, and you can read them in the PHB and DMG.
Those rules don't say anything other than GM decides. If the GM is having a conversation with their players about what constitutes oathbreaking, then that's the GM using their authority to apply both constraints and operationalizing oathbreaking.
I'm not suggesting anything that the rules don't do themselves.
You're implying the rules are much more robust than they are. The "rules" for oathbreaking are weak and don't provide the player with handles except that it's up to the GM. This makes the rules for oathbreaking entirely a matter of trusting your GM. Again, not necessarily a bad thing, but it's fairly trivial to find horror stories on this exact topic that are both, well, horror stories and entirely within the rules as presented. IE, the GM is granted this authority with no constraints so even when outcomes might be less than desirable, their still within the rules. Suggesting the oathbreaking rules are more than the assignment of authority without constraint is adding things. Use of that authority is use of that authority, which is as I said, the GM adding constraints to the player (presuming the GM bothers to discuss it at all).
Now, I generally admonish others by going to bad faith play, so I'm going to admonish myself, here, and note that in good faith play the oathbreaking "rules" are usually sufficient. This assumes good faith between the player and the GM, so the outcome of "GM decides" should be clear and follow from the fiction and no one should be surprised. As I noted earlier, not having things tightly constrained or operationalized is okay -- 5e does just fine doing this quite a lot. And, a number of people like it this way. Some don't. Either way, it's good to be critically open about what's actually happening in play, even when you get good outcomes (because you're a well adjusted adult person playing with other well-adjusted adult people and not being jerks to each other, usually). In the case of the rules for oathbreaking, there are none outside a blanket reassignment of authority from the player to the GM to determine the PC build effects of player choices. And, this is fine.