My client wants a "show all" option for search results. What to do?

I've recently released a major update to a clients site. The site has basically been re-written from the ground up to be much more efficient, and hopefully much more functional.

The old site, however, had an option that the new site doesn't have, to display all search results on a single page. The new site doesn't have this because it has pagination, something the old site couldn't do.

From a technical perspective, the "show all" functionality of the old site was something that I had always been keen to fix. I had always considered it a fault, because it made the pages very heavy, and put a lot of unnecessary load on the web-server, database etc.

My client though is concerned now that the old site had a "feature" which the new site doesn't.

While I 'could' implement a 'show all'. I just really really do not want to. It just feels wrong to me. It feels like all the work we've done to optimize the site is going to be essentially thrown out the window, to implement a bad feature.

I obviously can't just say 'no' to my client though.

Without going into specifics, most of the results in a default search would not be relevant to any individual user. Due to the nature of the site, any individual user is never going to need to see every item in the list, and realistically would only ever need to see maybe 5 - 10 items.

I think users really should be using the search filtering, rather than idly scrolling through hundreds of irrelevant listings to find something that interests them.

What I'm hoping for, is to find some convincing arguments for why we shouldn't allow users to "show all", and what we should do from a ux perspective, so users would never want to "show all"