I believe the left auxiliary radiator is used with a 7-speed DSG transmission.
The right auxiliary radiator provides additional cooling for the main radiator.
Guessing a manual transmission TTRS's might not have a left aux cooler?

Personally wouldn't touch a rebuilt title except maybe for parts.

Unlike the UK, MK2's are relatively rare in the US, TTRS's even more so.
The total number of MK2's sold in the US (all years, all versions) is less than the number of TT's sold in one year in the UK (also consider the size of the UK compared to the US...)

Over here used TT parts are hard to find and dealer pricing (if available) can be ridiculous. If the TTRS in question was written off by insurance good chance it'll be a money pit.

Edit - Adding this:
P/n 8J0121212A “Additional cooler for coolant - left”
Applicable PR code G1D = 7-speed automatic transmission for four-wheel drive
See: https://www.lllparts.co.uk/catalogs/audi/USA/TTRS/676/1/121/121070

 

Yeah it's to cool the circuit for the DSG cooler which US cars could never have for Mk2s. So it's not missing it just wasn't built with it. The one on the right side is probably there though, it cools the circuit for the engine oil cooler.

 

Having recently pulled the front of my TTRS, I can confirm that there's no aux radiator on the left - only the right.