It's not going to be possible to get something that fits everything (speaking realistically). You would have to use EQ and unfortunately, iOS is pretty bad for dealing with EQ compared to Android. As it seems like you want an emphasis on bass and several of the genres would benefit from a V shaped sound, consider the Sennheiser IE300 if you are fine with IEMs. They would have a heavy emphasis on bass and treble. If you want a more balanced option with similar bass, consider the Sony XBA N3 as it has a "neutral with a bass boost" sound.

If you want a visual representation of the difference, look here:

https://crinacle.com/graphs/headphones/akg-k371/

https://crinacle.com/graphs/iems/graphtool/?share=IEF_Neutral_Target,IE300,N3AP

For over ear headphones, you could consider some wireless headphones as they would put more emphasis on the bass with the ability to adjust the sound on the apps (such as the Sony XM3/4/5, Momentum 4, Momentum 3, etc.).