Is it normal for a display board to run warm inside a wooden case?

That cable notch alone won't move enough air. Heat rises, so with only one small opening at the bottom you've got nowhere for the warm air to go, it just sits and the whole cavity soaks into the walnut. That's why the back feels uniformly warm instead of just around the board. Wood is a decent insulator, wich is great for looks and bad for cooling.

Warm to the touch after an hour is pretty normal for those driver boards, they run hot by design. The number I go by is the finger test: if you can hold your fingertip on the board itself for a few seconds without flinching, you're fine. If it stings, you need more airflow. Sounds like you're in the first category, but a sealed box has no headroom, so it'll creep warmer in summer or during a long movie session.

If you want vents, you need a pair of openings, not just one. A couple of slots near the top of the back panel lets the warm air escape, and the cable notch down low becomes your intake. Convection does the rest with no fan noise. Slots on the top edge facing up will also collect dust and the occasional coffee splash, so I'd put them on the back face near the top instead. Three or four narrow slots looks intentional if you space them evenly, almost like a design detail. If you really don't want to touch the look, another option is sticking a couple of small adhesive copper shims or a thin aluminum strip between the board and the wood to spread the heat out. Doesn't remove the heat but stops any one spot from getting toasty. Honestly though, for a second monitor that's on a few hours a day, two discreet slots in the back is the cheap insurance route. Nobody sees the back of a desk display anyway, so that's where I'd hide them.

I totally agree that you need more than just one opening. When I did a similar build, I found that instead of just cutting slots, routing a small channel inside the case right under the board was way better for airflow. It directs the air flow much more effectively than just relying on exterior gaps. Also, make sure whatever mounting bracket you use has a good thermal pad between the board and the walnut so it can pull heat away from one spot.

I dealt with similar heat issues when building a small rack for a modded PC, and I found that airflow is only half the battle. The board itself often just radiates heat right into the enclosure. I actually used thin graphite thermal pads between the circuit board and the frame. It sounded like a niche thing, but it made a huge difference. It helped transfer the heat away from the board and onto the wood, which then dissipated into the surrounding air before it had a chance to soak into the walnut itself. It stopped the whole area from feeling uniformly warm.