"The best workaround to include sortable columns here is to remove all the Query Filters, filter controls, pagination controls, and loop contents from this WordPress Archive and construct a similar View of Air Filters with the sortable table headers, front-end filters, pagination, etc. you want to show in the archive."
I have tried to do that, but can't get it to work.
I have created this view: hidden link
Where you inserted the View shortcode is inside the loop tags of the archive, meaning it will be output for every result of the archive.
Even if you don't want the archive itself to output anything, it still queries posts and iterates over the results, you just don't want to output anything from that query, as you are displaying something else.
So it's not a bad idea to update the archive settings to, say, a limit of 1, so there it is not needlessly iterating over results it won't output, and then move the wpv-view shortcode outside of the loop, like so:
Actually, you are right, the option isn't there on archives (only Views), but what you could do instead is set the pagination to display only one result at a time, which would have the equivalent effect.