Starfleet and the Federation has always been a prime factor in Star Trek. To have a trek TV show that had no Starfleet personnel is too far outside the scope of Trek, I think.
We have seen examples of non-Starfleet investigation teams before.
Seven of Nine's parent aboard the Raven.
Professor Robert Crater and his wife, in Man Trap, seemed to be a independent archaeologists.
Janice Lester didn't appear to be working for Starfleet.
John Gill was Federation, but not Starfleet. The same with Worf's brother, however a non-governmental team sponsored out of say a university could be doing similar research on a planet's culture.
Oh, don't get me wrong. I'm not saying the science teams would have to be Starfleet. But the crew operating the ship would be.
Basically, a Starfleet vessel that has science teams come on every few weeks, and not just science teams. You can have diplomatic teams to negotiate with a planet for Federation membership, a group of people starting a new colony, a team of doctors hurrying to a planet with a plague... The possibilities are endless.
But with a small group of core actors (the crew of the starship), you can have a group of other characters come on every few weeks, providing new story opportunities by giving the core cast new people to interact with. And you can also have crew rotations, transfers, deaths etc to open up spaces to have the starship crew move off the ship every once in a while and allow new crew members to come on.