You might find games that highlight the "Work in Progress" issue might help. Not only will they help to show the value of vertical slices, they also help with highlighting the issues of multi-tasking.
Here's a simple one that uses two common agile games in conjunction to train the concepts.
1- Production Line vs Finished Product
Give everyone a stack of papers. Then walk them through how to fold a basic paper airplane. Once the plane is done, each plane also needs to be numbered and given a window (using pen) where the cockpit should be. Once the how to instructions are done, then tell them that must do each step on all their expected workflow before moving to the next step. So if they think they can make ten airplanes, they have to fold all ten sheets of paper in half before going on the folding the left corner down.
Have them each guess how many airplanes they will get done in two minutes. Then let them loose.
I've never seen anyone get all their work product done on this. There will almost always be a lot of in process work.
2- The Assembly Line
Next you create an assembly line. Ideally seven people in the line. Each person has a single and specific action they do in the line (e.g. one person’s sole job is to take a piece of paper off the pile and fold it in half). Each person also has three spaces in front of them. The space to the left is their incoming work pile. The space to their right is their completed work pile. The space in the middle is their doing pile (Expectation being the first and last person). No one is ever allowed to have more than one item in their completed pile. Until the item in their completed pile has been moved to next person's Doing pile, they can't put something new in. In the first run through, they aren't allowed to change anything from these base rules. There is also a final person, who is the inspector. They inspect each plane for how much it looks like all the other ones made and checks their flight ability (got to have some fun).
This game teaches two key things. The first is that by limiting flow, you increase production and decrease waste. The other is that by letting someone focus, their quality goes up. The planes that come out of this will almost always be of higher quality than those in the first exercise.