Avoid two search modes (modern, classic) in Sharepoint Subscription Edition (SE)

We are using SharePoint Subscription Edition (SE) and seem to have two search experiences: modern and classic.

I want to avoid confusing users and make search behave consistently across the site.

Scenario 1: site search box uses modern search

When users search from the “Search this site” box outside a document library, SharePoint opens the modern search page with a URL like:

https://domain.org/sites/site/_layouts/15/search.aspx/siteall?q=test

If I change Site Settings > Search Settings, it has no effect on this behavior.

Question 1:

Is it possible to change the URL used by the built-in “Search this site” box, so it opens a different page?

Scenario 2: Search Settings works for classic results, but not for modern URL

When users search inside a document library and click “Expand search”, SharePoint uses the value from Search Settings.

However, if I configure “Send queries to a custom results page URL” and enter the modern search URL above, SharePoint does not use it correctly. It appends results.aspx?k= and the result is a 404.

Question 2:

Can Search Settings be used with a modern search URL such as: https://domain.org/sites/site/_layouts/15/search.aspx/siteall

without SharePoint appending results.aspx?k=?

I did not find any setting in Central Administration that seems to control either behavior.

My goal is not to customize the results layout (not possible in OOTB modern search page as far as I know), but to make users land on a single, consistent search experience.