How complete is the project? Having separate CPTs is OK, but one taxonomy across all CPTs is not so good, as WordPress does not handle well a taxonomy shared by more than one CPT.

So, Tags (as shipped with WordPress) are for Posts, but should not be used for other CPTs. It is possible to do, but not advised. Better to use a custom field (as can be made with Advanced Custom Fields plugin. Filtering is then possible across multiple (Custom) Post Types, as ACF allows field groups to be assigned to more than one Post Type. Sorting then becomes easier too.

Also, when our form is attached to an Elementor Post Grid, Elementor will apply this to the Post Types set in the form, not just ‘Posts’.

I really need to step you through the process as you build it, so start small and build up?

Try as I suggest with Elementor Post Grid, S&F and ACF and make a form and get this working first. Please see this post also:

https://support.searchandfilter.com/forums/topic/how-can-i-search-elemntor-column-while-retaining-the-design-of-that-column/#post-209754

Some time ago WordPress changed the way taxonomies (which include Tags and categories) are linked to Post Types, making searching across the same taxonomy in multiple different post types problematic. Be aware that this could cause difficulties. It might, for example, not find posts from all Post Types, or might count them wrong.

What would be much better, to enable me to help you as quickly and efficiently as possible, is for you to make a form, for you find one most important thing wrong, and then for me to look at what that is.