Reland "Use cast http connection in libassistant."

This is a reland of 36f90a0e10de3a0ac87791db9ccb463246e6e867

Original change's description:
> Use cast http connection in libassistant.
>
> Bug: b:118193766
> Test: Manual
>
> Change-Id: Ia0cd58e42761b35b9f7281344266505074d59936
> Reviewed-on: https://chromium-review.googlesource.com/c/1321956
> Commit-Queue: Yilei Li <yileili@google.com>
> Reviewed-by: Xiaohui Chen <xiaohuic@chromium.org>
> Cr-Commit-Position: refs/heads/master@{#607318}

Bug: b:118193766
Change-Id: Ice6e98bea5c1d97f538d7de98c4c799dc72788ae
Reviewed-on: https://chromium-review.googlesource.com/c/1337292
Reviewed-by: Xiaohui Chen <xiaohuic@chromium.org>
Commit-Queue: Yilei Li <yileili@google.com>
Cr-Commit-Position: refs/heads/master@{#610923}