Brahmāyu said this:
MN91:36.7: in that very seat the stainless, immaculate vision of the Dhamma arose in the brahmin Brahmāyu:
MN91:36.8: “yaṁ kiñci samudayadhammaṁ sabbaṁ taṁ nirodhadhamman”ti.
MN91:36.8: “Everything that has a beginning has an end.”
And then the Buddha said this:
MN91:39.6: “Mendicants, the brahmin Brahmāyu was astute. He practiced in line with the teachings, and did not trouble me about the teachings.
MN91:39.7: With the ending of the five lower fetters, he’s been reborn spontaneously and will become extinguished there, not liable to return from that world.”
So the Buddha would find no issue with that phrase.
Now the second question about whether beginning/end is the same as "All conditioned things are impermanent", is similar but requires another insight that points to a deeper wisdom:
SN22.81:5.5: And what’s the source, origin, birthplace, and inception of that conditioned phenomenon?
SN22.81:5.6: When an uneducated ordinary person is struck by feelings born of contact with ignorance, craving arises.
And even deeper:
MN1:172-194.26: Because he has understood that relishing is the root of suffering,
Understanding the impermanence of beginnings and endings provides a crucial basis for setting foot firmly on the Noble Eightfold Path to the end of suffering. And if a fake Buddha quote inspires one to read the actual teachings of the Buddha, isn't that beautiful?