Thanks for the appreciation!
Yeh, I threw my unstructured thoughts into Gemini and asked it to make them into a coherent and well-formatted answer for the forum, in my previous message. It ended up more verbose than I thought, but I didn't have time to refine it further. Apologize for that.
Now I'm writing myself to avoid further fun.
Now leaving the AI aside, CPM-3V is not used in axes because it is not forgeable and because other steels are better than it for the application.
Based on knifesteelnerds.com, by toughness, L6, 1085 and 80CrV2 are rated to an 8 out of 10 (probably based on a charpy or similar test), CPM-3V to a 9, while 5160 has a 9.5. S7 shock steel is not there, but I know it has a slightly lower impact resistance and slightly higher lateral strength than 5160, so it's probably also close to a 9.5.
Now here is the interesting point: Hardness also has a play in this and there are some nice charts with different steels at different hardnesses. CPM-3V at 58 HRC has around 41 ft-lbs toughness (i.e. resistance to chipping or breaking when impacted). At the same hardness of 58 HRC, L6 steel has 45 ft-lbs, while 5160 has a toughness of 45 ft-lbs at a hardness of 59.5 HRC!!!. That means that practically all these steels have better impact resistance than CPM-3V if used at around 58 HRC (the generally prefered hardness of high performing axes). The only advantages you'd have over these steels, if using CPM-3V in an axe, would be additional wear resistance and some extra corrosion resistance. The fact is that wear resistance is not valuable in an axe. Its main usage is to cut by striking. Even if you use it to do some feather sticks in a survival situation, that would still be done via push-cutting and not slice-cutting or friction-cutting. The shape of a hatchet is simply pushing you towards push-cutting and strike-cutting, not slice-cutting (as you do with a knife). So you get no extra-value by having CPM-3V there. CPM-3V is excellent if judged by its balance of wear resistance and impact resistance (which are highly valuable in a knife). If you need one of these properties more than the other, there are other steels that fit better. In applications like an axe or hatchet, where friction-cutting occurs rarely or not at all, what you need is extreme impact resistance at high hardness (i.e. combined with extreme deformation resistance). The optimal steels here are the ones I mentioned: 5160, S7, L6 followed by any carbon steel with a carbon content between 0.6% and 0.8% (especially if it has some extra small amounts of added silicon or nickel). The most appreciated axes in the world are using one of these steels.