Is there any way to get a list of all the URLs of all the ArcGIS REST servers that exist?
The answer here is much the same as the answer to "Is there ay way to get a list of the URLs of all the web servers that exist?" and for a similar reason. Anyone can create one with whatever host name they like, without having to register it on any index anywhere.
Using Google to search for the particular strings that are common to ArcGIS servers is an option to try, but of course it will not find them all.