Remove core::any::AnyPrivate

[Previously](https://github.com/rust-lang/rust/commit/e5da6a71a6a0b46dd3630fc8326e6d5906a1fde6),
the `Any` trait was split into a private portion and an (empty) public
portion, in order to hide the implementation strategy used for
downcasting. However, the [new
rules](https://github.com/rust-lang/rust/commit/e9ad12c0cae5c43ada6641c7dc840a0fbe5010a2)
for privacy forbid `AnyPrivate` from actually being private.

This patch thus reverts the introduction of `AnyPrivate`.

Although this is unlikely to break any real code, it removes a public
trait and is therefore a:

[breaking-change]
Revise DST test to not require duplicate errors.
auto merge of #17745 : aturon/rust/revert-any-private, r=alexcrichton

[Previously](https://github.com/rust-lang/rust/commit/e5da6a71a6a0b46dd3630fc8326e6d5906a1fde6), the `Any` trait was split into a private portion and an (empty) public portion, in order to hide the implementation strategy used for downcasting. However, the [new rules](https://github.com/rust-lang/rust/commit/e9ad12c0cae5c43ada6641c7dc840a0fbe5010a2) for privacy forbid `AnyPrivate` from actually being private.

This patch thus reverts the introduction of `AnyPrivate`.

Although this is unlikely to break any real code, it removes a public trait and is therefore a:

[breaking-change]