Wether or not you style the archives with Layouts, you will generally use Archives, not Views, and those are under Toolset > Archives.
In layouts, that's an Archive Cell.
A View is not used on archives, because an archive is basically already kind of a View, but restricted (by the code, not by Toolset), to the archive's main query object (posts, term, etc).
WordPress Archives, if made with Toolset, also support searches.
You can NOT split the search off the results, if you use a archive, that's right.
In the cases you need to split the search and the results only a View would help (with this feature you can display the search anywhere and results anywhere else).
You should generally not insert each of those elements twice.
What you can do is design your Search HTML so that it puts single inputs in different locations.
But inputting the same fitler twice just results in 2 identical filters be ing generated and of course, updating each other.
The way you achieved the goal is likely the only if you do not edit the HTML of the Filter section so to match the layout (which is not possilbe in this specific case, I see that)
It's however not best practice, you have now duplicated items on the template, in the HTML that is visible.
What you would need is the feature to build your search and results more freely, maybe GUteneberg (thhe beta Toolset Views) edit mode will help you in future, but right now, it's not possible to move around single items of a search like you require (unless you apply something similar to your solution)
In Gutenberg and Toolset Views' Beta right now you can edit already much more freely the filters, but such implementation still is not so easily possible as the filter still is as "one object" thing, rather than a "many objects" element that you can freely put on the template.
Please let me know if you need more details!
Thanks!