Here is my experience:
I left with the intent of getting permanent residency and decided to get the 6-month visa before I left. My reasoning was that if I already had it there will be no surprises if I tried getting one there. Things were changing rabidly at that time, so I was not sure the requirements would stay the same. You do have to get the 6-month visa registered in Ecuador, so don't forget to do that. In other words, the paperwork they give you is not the visa. You have to have your passport stamped in Ecuador.
One real benefit of getting the 6-month Visa in the U.S. was that the Ecuadorian Consulate gathered up their and my documentation and and stapled and stamped all the pages so I had a sealed and organized packet. The clerk did this because I forgot to get the translations certified. HIs seal apparently fixed that for me. The entire package was accepted here without comment, but there were some back and forth discussions regarding forms or paperwork which were not included in that Consulate package. That saved me from what could have been a real headache.
Like the gentleman said earlier: take everything you can think of (certified and with apostille, when required), although U.S. tax files probably aren't needed. One other caution, if a document or certification is over 6-months old, they can reject it. I had that same problem in the U.S. when I tried to get an apostille for some of my older certified documents. So, I brought everything up-to-date. But a friend did have one document rejected in Ecuador and had to get another. But, experiences vary so one example does not set the rule.
Finally, if you file for permanent residency while you are legally here, you will be able to stay until the status of the application is determined. You may have to have a Lawyer write a letter to enact this benefit (and so you will documentation to carry). I didn't - mine took 8 months - but this is something you should ask when you file for residency. Never hurts to ask as much as you can think of. There is always a good reason to ask. If you are given an answer, people tend to honor that. For example, if you ask what a taxi ride costs before you get in, then that will be the price. Enter the taxi without asking, and you could be surprised. For taxis, this mostly applies to people who are obvious tourist. If they think you are local, they will charge you the standard rate which everyone knows. These comments apply to Loja, I don't know about other cities.
Another example, I had a medical exam and asked the price. The wrote in down for me. When I went to pay, I was told another price. but the person noticed the paper (without my pointing it out), apologized, and charged me the written price.