I've played with a few VTC puzzles with a tetrahedral mechanism rather than octahedral. It simplifies the design but it's much harder to get the tension uniform on the puzzle and it's much easier for big groups of pieces to pop out. I recommend a octahedral core if it's feasible.
You do bring up a good point but there is actually a simple way around the popping problem. By using both symmetries of the tetrahedral mechanism in two separate layers, it becomes more stable.
I'll use the Skewb as an example. A regular Skewb has just one groove underneath the center pieces and half of the corner pieces. I've realized it's possible to add a second layer of grooves underneath the first one to make the pieces more stable, as you can see below:
File comment: Center
skewb center.jpg [ 177.51 KiB | Viewed 797 times ]
File comment: Floating corner
floating corner.jpg [ 163.16 KiB | Viewed 797 times ]
File comment: Anchored corner and core
anchored corner and core.jpg [ 160.56 KiB | Viewed 797 times ]
Here's an image of the type of interaction between pieces:
File comment: Interaction 1
interaction 1.jpg [ 177.73 KiB | Viewed 797 times ]
File comment: Interaction 2
interaction 2.jpg [ 174.78 KiB | Viewed 797 times ]
Keep in mind that it still uses a 4 armed core, but because all of the pieces are gripped evenly on all sides, the puzzle becomes (nearly) as stable as an octahedral core.