You could, but it's not really safe and takes a lot more work and you have to adjust it for each diode separately. And even then it will have a different current every time you turn it on.
With current regulation, you just select the correct resistor, and the current will always be exactly what you want it to be, regardless of small differences in Vf between the same type of diode or even with a completely different diode.
But these circuits are completely unregulated. At most, the peak switching current is limited. They drive a LED by "blinking" it very rapidly - there is no filtering.
You would have to filter it first with come ceramic SMD caps. Then you would have to select the correct resistor, that would limit the current to a level BELOW the max safe current for the LD, WHEN THE BATTERY IS FULL! This resistor would be different for every diode, because of a different Vf.
You would have to set it lower, because it would change. As the diode heats up, the current would climb. As the battery voltage drops, the current would drop.
This means, you would have to limit the current enough, that it would still be below the max safe current, even when the diode becomes very hot on a completely full battery. This is something you need to do with a PSU, set to the highest voltage a battery of that type can have. Otherwise, the battery voltage can start dropping, and you think you limited the current enough, but when you put a new battery in, the diode gets killed. The same can happen if you put a different battery type in.
In the end, you are left with a laser, that is underpowered most of the time, and the power is never the same twice, when you turn it on.
It's even less safe, than direct drive off a battery through a resistor. And everyone will tell you not to do that.
There is a reason we use constant current drivers. It's hidden in their name...