Absolutely all hunter-gatherers live "hand-to-mouth", malnutrition is common, and starvation is not just a "risk" - it is a permanent threat.
This should be obvious because they cannot effectively store excess food and thus are subject to the standard predator–prey model:
plenty of food -->
population expansion -->
depletion of food sources -->
population contraction -->
plenty of food
native societies were relatively very stable
Medieval inner city looks like a modern college campus "safe zone" compared to the level of violence in "native societies".
People could could call on a wide range of resources
This is an exaggeration: a tribe hunting deer will probably have little to fall back on, especially in winter.
A tribe is a single social unit. They hunt together, they eat together, they starve together. Remember, they cannot store food! Yes, some have a better teepee or moccasins, but not food.
A neighboring tribe might fare better (unlikely, but possibly), but the distance kills cooperation. They are not likely to be bosom buddies (they attack each other to kidnap women all the time), and transporting food is very hard.
These fared better, but not by much.
They did not have high-yield crops and draught animals (and thus the wheel).
The first meant that they still lived hand-to-mouth (even though sightly better than hunter-gatherers because grain store better than meat) and the second meant that a local crop failure (due to, e.g., a drought) could not be mitigated by import.
See Guns, Germs, and Steel for details.
Effective food storage is a relatively recent invention.
Pre-industrial societies did not produce much excess food for storage and could not store if very well.
E.g., GurvenKaplan2007 mention
the case of an Nunamiut Eskimo group that perished in its entirety, having been snowed in without sufficient food supplies to survive through the winter.
PS. I am not saying that if a man fails to catch his daily quota of fish/fowl/venison, then his family will starve tomorrow. They can survive on the combination of yesterday's catch and the wife's gathering.
Death from starvation was probably not an annual event.
However, hunger was.
PPS. Further reading: