I think the "hostage situation" came up as a rumored reason for 3rd party countries to issue passports, not as a desired reason by the OP.

Anyway, the issue is that certain countries charge high visa fees to US citizens, and the OP wishes to avoid this. If this were easy, everyone would do it.

You have to either find a country that will admit you as a citizen, or issues passports to non-citizens. The latter is rare, and is usually restricted to refugees.

Unless you do have the right relatives, getting citizenship in another country is going to cost a lot more than the visa fees. Renouncing your US citizenship won't help either; you need to get citizenship in another country that has lower visa fees. Having no passport at all won't help.