The HHR is only responsible for issuing ID’s…they don’t inspect Health certificates, and yes, you’re right, they probably had no idea what you’re talking about (TARC, “Jail sentence” etc). To get the Taiwan ID after being on a TARC for a year (or 2 or 5 years), what the HHR needs is a document called the “定居證 (Certificate of continuous residence in the ROC or as you put it, “the jail free paper” 
)”. This 定居證 is only issued by the NIA and they will issue it to you after you spent a year on a TARC and with a Health Check. Didn’t the NIA give you a piece of paper (after you received the TARC) with detailed instructions on what to do and what documents to collect when you come back after a year?
I have wrote in detail on this forum on the exact steps I took from DAY 1 till I got my ID. Although it was almost 4 years ago and I went the Naturalization route (parents were not Taiwanese), some of the steps are still the same while some major changes also took place (i.e one doesn’t have to renounce their original citizenship BEFORE becoming a National)…you can read it here—> How to get Taiwan Citizenship - Primer, FAQ, and Resources … If you don’t wanna read all that, you can skip directly to post no: 1232 How to get Taiwan Citizenship - Primer, FAQ, and Resources which applies to your case.