There is a degree to which this move is "less democratic", but we have to be careful. It is less democratic in the governance of the Democratic Party, but it's worth noting that the Democratic Party is not a polity. Running someone who can win instead of who the party chose might be a more democratic outcome with respect to the organization that matters more.
There's absolutely room for this to be taken too far. With mostly just the two parties, elites could (and arguably did, pre-'68) use these kinds of concerns as cover for excluding certain perspectives from power. That's something to watch for, but given the context it is obviously not what is happening here when the person being excluded is the current President, with the support of the party elites up to this point.