[feature-flags] Remove aws_types dependency

Summary:
Same as D12173 but for feature-flags. This is the last usage so removed it also from root Cargo.toml

Depends on D12173

Test Plan: feature-flags service compiles. Also did a global cargo check.

Reviewers: varun

Reviewed By: varun

Subscribers: ashoat, tomek

Differential Revision: https://phab.comm.dev/D12174