As described in #9576, the `SearchParamsProvider` can become stale when
navigating routes and relying on search params during initial render.
This is because this context, along with the `ParamsProvider`, is
duplicative to the internal lifecycle of `useSearchParams` and
`useParams` from `next/navigation`– but always one render behind.
Instead, we need to use the hooks directly from `next/navigation` as
described in the jsdocs. This will also remove any abstraction over top
the web standard for `URLSearchParams`.

For this reason, these providers and their corresponding hooks have been
marked with the deprecated flag and will continue to behave as they do
now, but will be removed in the next major release. This PR replaces all
internal reliance on these hooks with `next/navigation` as suggested,
except for the `useParams` hook, which was never used in the first
place.

```diff
'use client'
- import { useSearchParams } from '@payloadcms/ui'
+ import { useSearchParams } from 'next/navigation'
+ import { parseSearchParams } from '@payloadcms/ui'

export function MyClientComponent() {
- const { searchParams } = useSearchParams()
+ const searchParams = useSearchParams()
+ const parsedParams = parseSearchParams(searchParams)

  // ...
}
```
_MyClientComponent.tsx_

…er (#16884)

Removes the `ParamsProvider` and `SearchParamsProvider` from
`@payloadcms/ui`, which were deprecated in #9581 since
[v3.3.0](https://github.com/payloadcms/payload/releases/tag/v3.3.0).

The `useParams` and `useSearchParams` hooks remain available from
`@payloadcms/ui` with identical usage — they now originate from the new
[Router Adapter](#16763)
instead of these dedicated context providers. No migration steps are
needed.