You are absolutely correct in that the feat allows you to redirect the attack to yourself without any riders on reach. We can argue about how illogical that is, but that is one of the powers that the feat that you have invested in gives you.
Nevertheless, there is enough leeway in the interpretation of the rules that your DM is fully justified in setting some conditions. So I would always go along with the ruling of the DM, and I do not disagree with the ruling as made.
Nevertheless, I would like to give some hints as to how I would handle the situation as DM as a possible alternative that works for me. (You do what works for your table and for the enjoyment of your players.) As DM, I am the full master of the monsters I have placed in my playground for the benefit of my players. I am therefore already master of the meta-game and do not need to fall into rules traps. If my player has invested in the Mounted Combatant feat, I am not going to nerf it by adding my own conditions. Instead, I will not have my monsters target the mount in the first place if the rider is out of range, since I know that it would be an oversight for the rider NOT to redirect the attack in that case. Instead I just have the monster move to where the rider IS within range and then attack the rider. Then I do not need to make any awkward rulings.
This gets more interesting if, say, the mount blocks a hallway so that my monster is unable to move around it to hit the rider. In that case, I would just have my monster use the Tumble or Overrun action/bonus action from page 272 of the DMG, and then hit the rider from the back side.
On the other hand, if the mount is huge or larger so that I cannot reposition the monster to hit from the other side either, then as DM I have the power to define the mount as "suitably large" to use the "Climb Onto a Bigger Creature" option in the DMG.
Now if my monster is too large to climb onto a Huge or larger mount (and as DM I would let a Large monster attempt to climb a Huge mount, knocking the Medium or smaller rider out its way), then it is only one of a small handful of Huge or larger monsters with insufficient reach or a ranged weapon to hit the rider anyway without needing to climb up. I would therefore ask myself "Why did I put this odd creature into my playground?" I can just swap it for an alternative creature with reach or a ranged attack if this is going to be an issue.
These different approaches add some action and dynamism to the encounter and makes combat a lot more fun for the players. I try to build up the epic nature of the narrative. I would have my creatures move, tumble, overrun or climb the mount to dazzle my players with some action. This for me beats having my creature just stand there targeting the mount when the Mounted Combatant rider is out of range, forcing me, the mighty DM, to make an awkward and unnecessary ruling. I never have to put myself into that position.
Anyway, this is just my alternative to nerfing a feat. As always, you do what works for your table.