How plausible would it be to build and use an upper stage with little to no avionics and RCS, just an engine and fuel tanks? Would it be possible to use a satellite's/probe's RCS and guidance for the duration of a burn?
It's not only plausible, it has been done. On the ATK STAR series of solid upper stages, a reaction control system is optional.
However, the more common situation is to have the avionics and RCS for a rocket installed on its upper stage, so the first (and maybe second) stage is the one without RCS and avionics.
On the Ariane 5, for example, the avionics and RCS are integrated in a structure called the Vehicle Equipment Bay. On the Ariane 5G, the second stage (EPS) sits inside the VEB. On the Ariane 5 ECA, the VEB sits on top of the second stage (ESC-A), and the RCS has been moved from the VEB to the ESC-A.
There are three reasons not to have the payload control the entire rocket:
The rocket's RCS has been sized to control a 1000-ton rocket instead of a payload that weighs a few tons. Such an RCS is relatively heavy, so it makes sense to install it on the rocket instead of having it eat up space and weight on the payload.
You need an RCS and avionics on the upper stage to deorbit it after payload separation.
Putting the RCS and avionics on the satellite makes the satellite and rocket designs more dependent on each other. It becomes more difficult to switch a payload to a different launcher. It becomes more difficult to test and qualify the avionics and RCS (because it exists in a different environment for every launch, instead of being a constant). The rocket designer has to give information to the satellite designer (information of the type ITAR was designed to protect), meaning you get lots more paperwork and it becomes almost impossible to lunch satellites made in foreign countries.
There are actually two questions here: 1) Would it make sense to have [upper] stages controlled by the spacecraft being launched (a "dumb" booster)? and 2) Would it make sense for upper stages to have no directional control and just be an "inactive" propulsion system?
The first question's answer is yes - in fact, the Apollo Guidance Computer could be used to control the entire Saturn V stack, not just the upper stages. Adding a few more command sequences to an existing piece of hardware to more fully utilize its capabilities makes a lot of sense: Unless you end up having to add more memory, computer commands have zero mass.
On the other hand, while it would be plausible to build a completely "inactive" upper stage, it would not be reasonable to do so: In order to have the payload engines control the upper stage, you would have to add mass both for the maneuvering fuel and for beefier engines to have a effective control over the larger vehicle. This means your satellite/probe (launch payload) would be larger and more expensive than an "active" upper stage design would require. You would then either need a larger booster to carry the same mission payload into space, or reduce the size of the mission payload to compensate for the increased propulsion system overhead. In addition, an "active" upper stage isn't going all the way to the target orbit or trajectory, so when staging occurs and it's left behind, the extra mass of its control mechanisms is removed from the component being propelled toward the final destination.
However, RCS engines are not the only option for providing directional control for an upper stage: The stage's rocket motor can also be gimballed to control the direction of the thrust vector, and thereby the direction the vehicle is going. Using an RCS to point the stack while the upper stage engine is not firing would not require substantially larger RCS engines because the change of heading could be done more slowly, but to change direction while the booster engine is firing, gimballing the engine makes a lot more sense.