Wrong. You are conflating enforcement of the law - which obviously the police can do by arresting people - with the authority to execute unarmed people simply because they are in the commission of a crime. These are very different concepts. The police can use lethal force only in very limited circumstances even when a crime is being committed.
Well, there we have it. The martyr. Babbitt was "executed unarmed".
You don't have to go very far to find alt-right groups and media promoting Babbitt as a cause celebre. A few weeks ago, GOP Rep. Paul Gosar of Arizona (whose three siblings are trying to get their brother removed from office because he's a racist) said the Capitol police were "lying in wait" for Babbitt.
For these guys, getting their "news" and lamebrain ideas from Fox News and Breitbart, to name a few, isn't about about keeping informed or understanding -- it's about right-wingers feeding their addiction.
The police cannot execute people who jaywalk or speed. And then say they did so because the person was in the commission of a crime. How did that work out for the police officer who murdered George Floyd?
This is as unhinged and false an equivalency as can be imagined. Neither Floyd nor Babbitt were "executed". Babbitt was shot in the midst of trying to beech a barricade to the Speaker's Lobby, after threatening police and urging others on with "Go! Go!" The officer might have thought she had a gun in her hand; now, if that comes out, you'll say he's lying. In any event, he felt she posed an immediate threat to himself or others.