Move provider picker of Automator to use new form element

[Tracker]
Update Summary: [One-line status update for stakeholders]
Short Description: [One-line issue summary for stakeholders]
Check-in Date: MM/DD/YYYY
Metadata is used by the AI Tracker. Docs and additional fields here.
[/Tracker]

Problem/Motivation

We have a new provider picker in #3556181: Create a form element for selecting providers. We should make sure that the Automator uses this, instead of its current way of manually creating the picker, that is a lot of redundant code.

Proposed resolution

Use the MR as testing ground for #3556181: Create a form element for selecting providers and use the form element instead.