Since there can be planet-like astronomical objects that have larger masses than the Sun (for example an astronomical object with a large iron core), I'd think that there could be "Solar System"-type systems that are similar to our Solar System except that the center which the planets are orbiting around would not be a star. Is there a name for these kind of systems, or do they not exist for some reason?
A mass of iron equal to the mass of the sun would be compacted to the size of the Earth. This is close to what a white dwarf star is. Anything that massive would not be held in a large volume by the repulsive force of electronic shells of atoms. It would be compressed into the degenerate electron state that would exert a pressure to keep the ions from imploding further. A white dwarf star, or an aged one that has lost its heat and gone dark would correspond roughly to what you are talking about.
As Bob Bee points out, iron forms mostly in supernova events. The sun will enter into a helium burning phase at its end, and then onward up to maybe carbon. The sun will not likely not fuse elements above oxygen. It is then hard to know how iron would form a body in the way macro imagines. In the distant multi-tens of billions of years there will be more heavier elements around in nebula. Maybe there can then be the gravitational accretion of heavy elements into bodies without there being a star, This might happen now for all I know.
The system you are asking about is certainly possible but it would be wrong to call it a "solar" system since the hub is no longer an active sun or star. Planet size masses can certainly orbit a sun size mass. The planet's orbit is determined by the balance between gravitation and centripetal force. Whether the very massive body at the hub radiates energy or not is irrelevant.
In fact a star is born out of a huge sun sized mass only under a set of conditions.
It must be massive enough
The majority of the mass must be hydrogen, or proton plasma
I would posit that the majority of the large objects in the universe have not passed these 2 conditions to cause sustained nuclear fusion and thus ignite to form a star. So these “non-solar” systems would be the rule and solar systems being exceptions to the rule.
Such systems would be very difficult to find since, well, we cannot see them. They are dark. But it would explain some of the dark matter in the universe needed to explain galaxy rotation.