Why Benzene is a good solvent even while it is not polar
Benzene is a good solvent for many organic compounds, the term good solvent to me does not mean a nice solvent with a good all round profile.

To me the perfect solvent would be non toxic, not flammable but combustable, very resistant to radiation, inert to nitric acid, be easy to distill, be cheap and easy to obtain. It would also have to be able to dissolve a useful range of different things.

Benzene is more able to dissolve many organic substances than a saturated hydrocarbon such as hexane, this is becuase the pi cloud of the benzene ring is able to interact with the pi clouds of various solutes. For example polystyrene will be solvated better by benzene than by hexane for this reason.

This good ability to dissolve things is why I regard benzene as a good solvent. It is a good solvent to dissolve things but often a bad thing to work with in the lab. Benzene has one special use, as a NMR solvent. One of my favourite tricks is if my student can not understand a proton NMR spectrum due to overlapping peaks. I tell them to repeat the measurement in d6 benzene. Often a lot of the chemical shifts for protons change, with a little luck the resulting spectrum is more easy to understand. Sometimes you need to look at both the chloroform and benzene solution NMR spectra to be able to understand enough of the spectrum to understand what your structure is.

While benzene is an unpopular solvent due to the health issues and flammability, there are plenty of aromatic solvents which have better health and fire profiles. For example Solvesso 150ND is an aromatic kerosene which is marketed by Exxon, this has different solvent properties to the aliphatic saturated kerosene marketed by Statoil under the name of "solvent 70".

This can have some advantages, if I want to dilute the ionic liquid aliquat 336 to 30 % in a diluent and then use this solution to extract metals such as copper, cobalt and iron from aqueous chloride media. Then with solvent 70 as the diluent (diluting agent) then the organic phase is very prone to separate into a layer of ionic liquid with metal complexes as the anions and a diluent rich layer. But when an aromatic solvent such as ethyl benzene or solvesso 150 is used then the organic phase can hold a lot more metal before this unwanted phase splitting occurs.

The aromatic solvent (diluents) tend to be more toxic than the saturated aliphatic (alkane) diluents. If you look at "A comparison of two methods of recovering cobalt from a deep eutectic solvent: Implications for battery recycling" by F.J. Albler et. al., Journal of Cleaner Production, 2017, volume 167, pages 806-814 it contains an assessment of leaving a puddle on the floor of an unventillated room is made for both solvent 70 and ethyl benzene / toluene is made. The solvent 70 level in the air is likely to pose a far smaller threat to health than the aromatics. But aromatics like solvesso 150 are likely to pose a far smaller carcinogenic threat than benzene.

It is important to not regard all alkanes or aromatics as being the same, hexane is a popular solvent in chemistry but it is very toxic. In the body it is converted to hexane2.5-dione which is very bad for you. But the other longer alkanes do not have the same harmful effect. You also need to consider the exposure route. For example a puddle of diesel fuel on the floor of a garage is far less harmful than exposure of a person's hand to a high pressure spray of diesel which injects it into the person's body.