As I have already mentioned, Manusmriti, Skanda Purana, and Bhavishya Purana are some of the most corrupted texts in Hinduism. Brahmins made changes in Manusmriti and ganged up to protect the interpolations, because they had written these to benefit themselves. This is what Wikipedia has to say about Skanda Purana: "Skanda (81,100 verses), probably the longest of all, containing parables, legends and stories, with multiple versions and recensions. Many untraced quotes from a Purana are conveniently attributed to this Purana."
Bhavishya Purana, no doubt, is an old book, but it is one of the most interpolated. Here are some views on it from Wikipedia: "The Pratisargaparvan, though nominally mentioned in the Bhaviṣya (I.1.2-3), is practically a new work containing stories about Adam, Noah, Yākuta, Taimurlong, Nadir Shah, Akbar (the emperor of Delhi), Jayacandra, ... and many others. It even knows the British rule in India and names Calcutta and the Parliament." A. K. Ramanujan mentions finding references to Christ, Moses, and Queen Victoria in the "appropriately up-to-date Bhaviṣya Purāṇa" and cites this as an example of the fact that: "In spite of repeated efforts to impose schemes and canons on them from time to time, Purāṇas are open systems.
There is not much interpolation in the RigVeda, Upanishads, or BhagawadPurana. Hindus do not have a Torah, Bible, or Qur'an, which is binding on them (thank our stars for that). Any portion of Manusmriti, Skanda Purana, Bhavishya Purana, or any other book, which goes against common sense is summarily rejected. The passages that you mention are certainly there, but they stand only as a blot against hindus, and a reminder of dark periods. And they are far from the truth.
If the brahmins were not supposed to teach the Shudras, how is it that all the great books of hinduism were written by Shudras, Maha Rishi and Brahma Rishi Veda Vyas and Valmiki, and how come Vidur was considered the most learned person in Krishna's time? How come, Jabali, who was the son of of a woman in prostitution and a shudra to boot, is considred a sage? How come 7 out of 12 vaishnava saints of Taml Nadu are non-brahmins? How come hindus rever Kabir, Ravidas, Dadu, and even Abdur Rahim Khan-e-khana or Malik Mohammad Jayasi at par with Tulsi Das, Surdas, and other brahmin saints.