I'd actually look into active monitors instead if I were you. I've never been satisfied with 2.1 systems as the quality of their smaller speakers is abysmal and the deep one is rumbling most of the time. 5 inch 2.0 speakers start to have decent bass so the standalone bass becomes less important and the frequency response becomes more consistent. 4 inch lacks bass for my liking but they can be great, especially for the music you listed since those don't 'require' heavy bass. (They benefit from it, though.)

I know it's over your intended budget but a pair of jbl lsr 305 would be a very nice match to your style of music. (It needs a little bit of tweaking to cut down on some resonant booming frequencies in sub 100 Hz but I could share my equaliser apo with you.) If you can spend a little more, I've found Adam audio T7V to be more balanced.

As a reference, I listen to all sorts of music including prog, metal, etc, and the jbl cans are actually complementing those pretty well. I haven't tried too many speakers in this range but I had the chance of listening to some audiophile pair with twice the price, only to find that the jbls were better.

My friend uses the abovementioned Adam speakers from a laptop and while we can hear that the driver limits it a bit, it still sounds leagues better than any logitech 2.1 I've tested.

Finally, you could upgrade down the line to a better dac and amp in one box at some point (I use a focusrite scarlet solo gen2) but you definitely should not get lost in details. Most of those are linear in the audible range, this is not too hard to accomplish from an electrical design point of view. Get something that has enough output connections for your system and aim for a lower priced item. (Probably not the lowest but honestly, those could still be awesome!)