Also remember that search engines most often link to links than they (ever?) Fill out forms.
This problem, which allows scanners to see your db content, is called a dark web, an invisible network, a deep web, or a hidden network. Blog post
So your expression about the problem:
an aspectual search for a simple app for a shoe store that allows you to filter the color, brand and type of shoe and sort by price and size or something else.
I would suggest that you include a search in the link hierarchy in addition to searching through forms with selection fields.
For example, in the additional menu there are all different brands as separate links. Then each link should lead to a list of products sold by this brand. The trick is to arrange things so that the link to individual shoes returns you to the first page (an application with a rich one page), but shows specific shoes. - And the page should implement the Google Ajax workarounds that you link to in OP.
Larry k
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