"Vintage" products

Follow dialabrain's advice about doing an EtreCheck report. There may be something clogging the system.

A 2013 iMac while nearly vintage is still plenty fast and handle's Catalina just fine. I'm typing on 1 now. However, its standard rotational drive is a huge bottleneck. For a relatively small price you can purchase a high quality USB 3 enclosure (check out macsales.com for the Mercury Elite Pro enclosure) and add a 256GB or larger SSD from the same site or Crucial Tech. You can use TimeMachine to restore your operating system and applications to the external drive - depending on how much data you have and what size SSD you purchase you could move everything to the SSD. That's what I did with the iMac I'm using. I now use the iMac's internal drive as my TM drive.

Technically a Late 2013 iMac should not be considered "Vintage" (barely). Not sure what you can do other than try an Apple Store or a different AASP.

Vintage and obsolete products - Apple Support

You should post an EtreCheck report for analysis.

http://etrecheck.com/

Use the Additional Text icon under the reply window to paste in the report.

"Late 2013" iMacs were not discontinued until Oct 2015 so, per the Apple document dialabrain linked, Apple says that, at the minimum, your computer has Apple parts and service support until September 2020. In some regions, local consumer laws may extend this. Other service providers many not honor this.