While Stark did call JARVIS a "souped-up SIRI," I don't think he was being honest. JARVIS was a unique artificial intelligence created by, arguably, the smartest man on Earth. I believe he meant it in a comparative manner, looking at JARVIS in relationship to the alien AI software resident within the Mind Gem.
Can your SIRI do this? I didn't think so.
In the scene where he shows JARVIS in comparison with the software within the Mind Gem, he was comparing JARVIS with the "intelligence software," a far more advanced and completely neural-appearing operating system. JARVIS was likely the most sophisticated AI on Marvel Earth before Stark's attempt at the Ultron AI.
JARVIS had control of the Iron Legion, maintained all of Stark Enterprises private security, maintained his home defense protocols, ran programs and software integrations in Stark's stead and was able to manipulate and control all of Stark's Iron Men armors, and all of this in real time. JARVIS was also able to even in a semi-disrupted state, upload himself to the internet, hide and still defend the world's security systems and nuclear defense systems from a superior, but still learning, AI housed in Ultron.
I point these out because I believe there would likely not have been a full back up of JARVIS anywhere. He would have been too unique to easily backup. Given that Tony went to a different interface system when the events in Age of Ultron were over, lends credence to the idea that all that was JARVIS was destroyed and then reintegrated inside of the Vision synthezoid's computerized artificial intelligence which was partially a program derived from the Mind Gem, Ultron and the remnants of JARVIS.
JARVIS assisted Dr. Bruce Banner and Tony Stark in using the Scepter's "A.I." to finish the peacemaking program codenamed the "Ultron Program". The two scientists left JARVIS alone to upload the A.I. to join a party that the Avengers were having. Afterwards, Ultron awakened and quickly realized its situation.
Here is the bone of contention:
Before JARVIS could contact Banner and Stark, Ultron shut him down, effectively "killing" JARVIS. Secretly, JARVIS went into hiding by spreading its own memory around to avoid detection. When the synthetic body arrived, JARVIS was uploaded into it, becoming an entirely new being: Vision.
Realistically, it makes no sense for there to have been no copies of JARVIS anywhere, given Stark's incredible resources, so our options are:
The Ultron AI attacked (and killed/destroyed/decompiled) not just the instance of JARVIS running in the lab, but every instance of JARVIS running anywhere, preventing him from being able to run on any device capable of contacting Stark. This meant the fragments of JARVIS that ran into the Internet were ALL the JARVIS that was left. JARVIS would effectively be DEAD.
Or perhaps he may have also decided that JARVIS, now housed in the Vision, might be uncomfortable with another instance of itself running around. For Stark, the activation of FRIDAY was his way of paying respect to his old friend/interface and honoring his new relationship with the Vision.