If it would be practical, our ancestors probably would have done it. But they haven't. What stopped them?
Mammoths and elepants need to eat several kilograms of grass and other plant material each day to survive. That forces the people to move constantly in a nomadic lifestyle and it sets an upper number of individuals in a sustainable herd.
Your humans are supposed to eat mainly plant matter as well, which makes them direct competitors to the mammoths. It's unrealistic to let a mammoth pull a wooden contraption loaded with berries and other foods and expect it to never reach its trunk in there and eat from the supplies.
Your solution to this food competition was to let the humans drink / eat the mammoth milk or dairy products. At first glance this sounds like an excellent idea. The milk is probably rich in fat and proteins because the mammoth babies have to grow fast before the next winter strikes. A study about the composition of elephant milk found that elephant milk is 80% water and 10% fat and protein respectively. That's much better than the 3.2% and 3.9% fat and protein respectively of cow's milk, but not enough to feed an entire nomadic clan.
Elephants gestate for an entire year and lactate for 3 years (if I understand Wikipedia correctly). Additionally, one elephant bull mates with all fertile elephant cows of a herd. That means you get a stable milk supply from several cows for 2 - 3 years before you have to wait an entire year (or longer) for fresh milk (because most of them become pregnant at the same time). I couldn't find sources about the quantitive amount of milk produced by elephants, but you have to give most of it to the calf for at least half a year before you can take the bigger share for yourself.
Unlike in a cow's herd, your nomads would rely on a wild mammoth bull to impregnate their mammoth cows, which poses a great problem.
Bulls engage in a behaviour known as mate-guarding, where they follow oestrous females and defend them from other males. Most mate-guarding is done by musth males, and females actively seek to be guarded by them, particularly older ones.
Elephant males have a hormonal cycle like the females. "Musth males" means that the male produces lots of hormones and pheromones that attract the females and make them very agressive. Maybe the nomads could substitute for the guarding male, but that would lower the chances of pregnancy, because the guarding male would mate with the females several times. Or they would somehow eliminate the agressiveness in centuries of selective breeding.
And last but not least, let's not forget how very intelligent and social elephants are. If the nomads slaughtered an old animal of their herd, there's no chance the remaining individuals wouldn't notice and fear or even fight their herders. There's a chance to avoid such problems if the heard neither sees nor hears the animal that's being slaughtered, but given the size, thick skin and loud voice of elephants, I'm not sure how the nomads could pull it off.
It's suggested that stone age hunters chased mammoths to make them fall over cliffs to kill them. That is in no way a clean death. The animal is likely still alive, panicked and screaming. The herd would hear this several miles away.
To sum it up:
- The cost (plant material to feed the herd) is too high for the yield (the milk)
- Reproduction takes too long and involves bulls in an agressive mood
- Slaughtering would have devastating effects on the remaining herd.