iPad Air 5th Gen Heating Issue even when Apple Pencil is just attached

As described, your iPad or Pencil may have a hardware fault.

You would be best advised to visit the Genius Bar at your local Apple Store or Apple Authorised Service Provider (AASP) - where the technicians will be able to perform hands-on test and diagnostics for your iPad and Pencil:

Genius Bar Reservation and Apple Support Options - Apple

Alternatively, might seek assistance directly from from Apple Support. You can contact the Support Team using the Support link at bottom-left of this page. You can also initiate and manage your support cases using the excellent Apple Support App. If not already installed on your iPad, the App can be downloaded from the App Store:

https://apps.apple.com/gb/app/apple-support/id1130498044

The Lightning controller is an internal hardware component - present only on iPad models that have Lightning Port. The information that you have received suggests that repair will be required.

If your iPad uses a second generation Apple Pencil, the iPad doesn’t have a Lightning port…

Indeed, the information that you received is indeed somewhat strange.

Accepting that it is very unlikely that you were unlikely to have been purposefully mislead, perhaps a benevolent explanation might be that the agent - with whom you spoke - was referring to internal hardware by the incorrect term.

As suggested in my initial reply, you would be best advised to visit the Genius Bar at your local Apple Store or AASP - for hands-on testing and evaluation:

Genius Bar Reservation and Apple Support Options - Apple