You have referenced the rules correctly.
From the spell (emphasis added):
You cause a creature or an object you can see within range to grow larger or smaller for the duration.
From the DMG:
For the purpose of these rules, an object is a discrete, inanimate item like a window, door, sword, book, table, chair, or stone, not a building or a vehicle that is composed of many other objects.
There is nothing in the rules that says this won't work.
If you are a player
You can anticipate gleefully running through dungeons, banks, palaces, reducing doors as you go. Don't stop there. You can enlarge them, too, and see what happens. Reduce the stones in the walls of the castle and watch it fall, the wheels on the wagon and watch it crash, any number of other things. Reduce the throne under the queen, the tree the bird is perched in, the witch's caldron, the possibilities are endless.
Only you might want to work it out with your DM ahead of time. You can decide something will work a certain way by reading the rules, but you only really know how it will work when you try it in the game and the DM narrates the results of your actions.
If you are a DM
If you're happy with this use of enlarge/reduce, go for it.
If not, don't. Allowing enlarge/reduce to open any door may be contrary to what you want in the game. If it's an issue, feel free to customize it.
Customizing the rules for your game is not straying from the rules. Those are the rules. The DM is the final arbiter of how things work in the game.
I have not dealt with this particular rules challenge as a DM, although I've dealt with similar ones. A reasonable and simple modification is to require a skill check whenever an object is being enlarged or reduced. Most objects, set the DC so low as you not even need to check it. For objects that have some resistance, set a higher DC.
You could also simply rule that reduce/enlarge does not work on attached objects.
Depending on your specific game and players, you might find you need to fine tune your house rule to make sure it works for your table.
Some players may want to argue with you about it. I refer you to "Rules Discussions" in the DMG:
You might need to set a policy on rules discussions at the table. Some groups don’t mind putting the game on hold while they hash out different interpretations of a rule. Others prefer to let the DM make a call and continue with the action.
Usually it's less fun to have lengthy rules discussions in session. I find it's better to make a ruling and move on.