I don't do vertical ice and have carried an ice axe many times for stablity and in case the conditions iced up and what is typically a pretty benign, soft, slope became nasty and I wanted self-arrest abilities. As such, for me, an axe would need to suport several times my weight because it might have those loads during a self arrest. But it doesn't need to support a few tons - I'm not using ropes, no one is going to take a leader fall on it and I'm not preparing to potentially arrest a whole rope team. So, for me, I would go quite lightweight.
I saw in a previous thread someone pointing out that if you have to chop steps, them some weight in the axe head helps you do that. I can see that point, but I don't plan to cut steps and if I did, it would be 50 or 100 to pass a snow-filled gully, not thousands of steps.
Also, consider trekking/ski poles with self-arrest grips. I remember the Patagonia catalog stating you weren't supposed to use them to claw your way to the front of the lift line.
And whatever you carry, practice, practice, practice. I only ever once had to REALLY self arrest, and all the training from a 1-day RMI class and various times I practiced since, snapped in and I managed to self arrest in nasty conditions and avoided a far nastier slope below.
Hike your own hike.
Carry your own axe.