> But I assume (am afraid) that there is no scenario
I can use the search function in a (offline) download of a website!? 

This looks like  a dynamically generated page and those are not supported by
HTtrack. And Httrack will not download content from another domain by default.
So if the main website domain is discogs.com, and the database is stored on
discogsdb.com, this won't work for you.

Httrack only downloads static pages. 

> “Rate limit reached. Error 429. You have made too many http requests.
Please try again shortly. 

Here the sysadmin of discogs.com has made it a limit how many times you grab
any webpage from their site. So you cannot download it. Httrack does not have
the ability to pause N seconds between each file downloaded to even try to get
around this (but I just requested this feature minutes ago). 

The admins of discogs.com want you to go to the site so ads will load and they
will get paid. Running a website is very expensive and time-consuming,
especially if the programmers are making a custom website. (I'm  a
programmer.)

You didn't do anything wrong, you just found the limits of Httrack.