For context, the envoy's official webpage is https://www.state.gov/p/eur/rt/hlcst/. I learned of this special envoy because Rex Tillerson recently ended a number of other special envoys, but decided not to end this one (see http://www.cnn.com/2017/08/28/politics/tillerson-state-dept-envoys/index.html). Note that I'm not asking about Tillerson's specific motivations for not closing this envoy, but more generally about how this office survived up until the start of the 20 year rule.
So, this office hasn't survived for 70 years but was created in 1990 and exists because issues in relation with the Holocaust both on a legal and political level are immediately relevant to issues of US foreign policy and bilateral relations between the US and other countries.
The two specific issues that lead the office of the Special Envoy being founded are the issues of outstanding reparations, restitution, and legal payments or Eastern European countries after the fall of the wall in 1989 and a number of class action lawsuits filed in the United States against Swiss Banks in 1998 over the restitution of Holocaust-era assets (namely dormant Swiss bank accounts of Jews killed in the Holocaust, Nazi bank accounts full with stolen money, and outstanding insurance policies that the Swiss banks did not pay out because of their burden of ownership proof).
Th central problem was that when the World Jewish Conference's Claims Conference negotiated with Swiss banks, the latter were not very forthcoming because they insisted on death certificates, something that usually simply didn't exist for Holocaust victims. When the WJC filed a class action lawsuit in New York in 1998, eventually the office of the Special Envoy, which had been created with similar conflicts between claimants and Eastern European nations in mind, sprang into action because there was quite some political support from the US for the WJC in these claims and the US had no interest of upsetting US-Swiss relations, especially in connection to the issue of Swiss Banks, too much.
The office was crucial in negotiating a 2000 settlement, which created a fund that as of 2015 disbursed $1.28 billion USD to 457,100 claimants.
Additionally, the Office was crucial in assisting the restitution of stolen Holocaust era assets, namely works of art that had been taken from the owners by the Nazi government and had in the meantime found their way into various museums and private collections. It helped negotiate a 2009 resolution, in which 46 countries agreed to the principle of compensating Holocaust victims and their heirs for property lost in their respective countries and that restitution should be handled by the governments involved. This resultion however, is morally rather than legally binding.
Furthermore, the Office of the Special Envoy represents the United States on the Task Force for International Cooperation on Holocaust Remembrance, Education, and Research and has as part of this been a crucial diplomatic player in helping negotiate such things as the archives of the International Tracing Service in Bad Aarolson or of the UN War Crimes Commission become publicly available in recent years, as well as shape international privacy protection laws concerning public availability of information on victims and perpetrators of the Holocaust, or finance a variety of research programs on the Holocaust in various countries.
There are a variety of legal and political issues still connected to the Holocaust that are immediately relevant to US foreign relations and policy and hence this offices has existed for the last 30 years.