Seems sensible... though I'd prefer not having a new icon added to the toolbar. There's already the "Profiles" button I have no use for, but can't remove. I don't want a non-removable search-provider one added too!
Seeing as when you type in the Address Bar (or when it is blank) there is a magnifying-glass icon to the left, which current does nothing when clicked, why not repurpose that instead?
Have the list pop-up below that, as you've suggested, of providers the user already has configured inside Edge, then at the bottom add a "Manage search engines..." link that when clicked opens a new tab and goes to edge://settings/searchEngines