This is basically a long comment, so I'm making it CW. I just wanted to advertise some related work, because I wasn't sure if you were aware of it or not. Back in 2004, Dugger, Hollander, and Isaksen published a paper that showed how to modify the model structure on simplicial presheaves so that equivalences are detected on hypercovers (including Cech resolutions), and this works for very general sites (including étale). You can also do $\mathbb{A}^1$-localization in these settings. I don't know what the words in your approximations (1) and (2) mean, but maybe this does what you want in your paragraph:

For example, I want the map $X\times \mathbb{A}^1 \to X$ to induce an equivalence in this sense on étale sites in characteristic $0$, and for this to imply equivalence of étale cohomology and other étale invariants.

Isaksen also has a paper on étale homotopy theory from the point of view of pro-spaces. And Kirsten Wickelgren has recent work on étale homotopy theory that you might not have seen yet: 1, 2, 3. Hope this helps!