you ask an excellent question.
So what does, "all beings have Buddha nature mean" in the Mahayana in context of correctly practicing anityata, anatmata, dukhata and shunyata?
In order to clarify the Mahayana view further on the Tathagatha Garbha, here from the Vajracheddika:
The Buddha said to Subhuti, "This is how the bodhisattva mahasattvas rnaster their thinking. 'However many species of living beings there are--whether born from eggs, from the womb, from moisture, or spontaneously; whether they have form or do not have form; whether they have perceptions or do not have perceptions; or whether it cannot be said of them that they have perceptions or that they do not have perceptions, we must lead all these beings to the ultimate nirvana so that they can be liberated. And when this innumerable, immeasurable, infinite number of beings has become liberated, we do not, in truth, think that a single being has been liberated "Why is this so? If, Subhuti, a bodhisattva holds on to the idea that a self, a person, a living being, or a life span exists, that person is not an authentic bodhisattva."
If someone holds a view that "I have Buddha nature" or "all beings have Buddha nature", they are wrong, why? How can beings be in possession of Buddha Dhatu, that is free from the idea of being, I, me or mine? As soon as someone makes up the idea, "let me save all beings" -- they don't understand.
A Bodhisatva, because having practiced correctly and understands correctly, doesn't think there is an "I" who actually saves or "other being" to be saved -- that is the person who knows Buddha Dhatu in truth. Only the one who knows Buddha Dhatu can reveal it.
The above Nikaya passage shows Buddha revealing the path to Buddha Dhatu -- essentially without Buddha, this Buddha nature cannot be attained to -- which is why Buddha is the Guide and gives the Marga -- the path to Buddha Nature
which releases the moon from a cloud (reveals the Tathagatha Garbha)
How can anyone be said to be in possession of a hidden treasure that he can't find, understand or use in any way? Without this map or path given by Buddha, beings can dig all over the place for Buddha nature and never find it.