[Experiment] Remove "Load Latest" button (#7120)

* Remove "show latest" behind the gate

* Add HomeBadgeProvider

* Update provider state from home feed tabs

* Add Home badge to native

* Add Home badge to mobile web

* Add Home badge to desktop web

src/state/home-badge.tsx

@@ -0,0 +1,24 @@
+import React from 'react'
+
+type StateContext = boolean
+type ApiContext = (hasNew: boolean) => void
+
+const stateContext = React.createContext<StateContext>(false)
+const apiContext = React.createContext<ApiContext>((_: boolean) => {})
+
+export function Provider({children}: React.PropsWithChildren<{}>) {
+  const [state, setState] = React.useState(false)
+  return (
+    <stateContext.Provider value={state}>
+      <apiContext.Provider value={setState}>{children}</apiContext.Provider>
+    </stateContext.Provider>
+  )
+}
+
+export function useHomeBadge() {
+  return React.useContext(stateContext)
+}
+
+export function useSetHomeBadge() {
+  return React.useContext(apiContext)
+}