AN IRULE THAT ADDS A STRING TO HTTP REQUEST OR RESPONSE (POOL MEMBER ALONE DOESN'T LOAD)

Actually, and just to use the correct wording, you will not append a string to a pool member because selecting a pool member is a load balancing decision, then comes the http protocol manipulation which is done at layer 7 either on the client side or server side connections, examples are redirection and uri rewriting. So, the correct expression is to append a string to a http request or response.

And to answer the question, yes a redirection iRule will do the job, assuming you have a http profile associated to the VS.

However, your condition [HTTP::uri] starts_with "/" is always true so you will end with a redirection loop. Try with [HTTP::uri] equals "/"