"Should" not depend on style of music is right, theoretically. But in practice, since all turntables have failings in different ways, ways that are related to certain aspects of music and/or sound, it may be advisable to find a table that least harms music in ways most important to oneself or to the music one most listens to.
For instance, just as an intellectual exercise: organ music is better served by a table great at reproducing low frequencies than one weak in that regard, whereas music without a lot of deep bass but with long-held piano noted needs a table with the lowest wow & flutter, a failing less harmful to the organ music with no long-held notes. The table that provides great bass could be weak in regards to wow & flutter, making it fine for the organ lover, not so good for the piano lover. Of course we all want a table with no musical or sonic failings; how many of us has one?
But, really, how many of us has the luxury of any such choice? The cartridge is normally the first part the LP player chosen, for the same reason the loudspeaker is at the other end: it's a transducer, far more variable in sound characteristics than the other parts (uh, right?). Who picks his arm first, then his cartridge? Or his amp first, then for a speaker it works well with? A knucklehead. Cartridge chosen, an arm appropriate for the cartridge is next. Then, lastly, the table. Owners of Oracle Delphi's learned why that table was not the best choice for the ET air-bearing arm: the mass of the arm tube was too great for the table's suspension, changing the level of the table as it moved across the LP. The VPI HW-19, on the other hand, was a good match for the arm.