Are EL34 based amps more musical than KT series based ? Or it's purely a matter of design?

IMO purely a matter of preference. I have sets of both, but generally lean towards the EL34 sound… finesse over power 😉…

There's a bunch of variables involved.  It's a matter of preference, the brand and age of the tubes, the unique circuit it's playing in, synergy with the system, and a whole bunch of other stuff.....definitely not an absolute.  It's also one of things i love about using tubes....try a new tube, and discover a bunch of new subtle tonal characteristics. 

That's probably why I kept my Quicksilver Mid Monos.    They ship with EL34 but can take any EL34 or 6L6 variant.    I've used KT 66,77,88 but I lean more toward the EL34 so I will put them in my system occasionally.

I have a 300b too and prefer that but sometimes I'll mix it up and put the QS in 

It is a matter of taste and specific implementation.  While there are some rough generalizations one can draw about the sound of particular tubes, I don’t see such generalization extending to the KT label.  I generally don’t like the KT88 tube, but I do like most amps employing the KT66 tube.  That tube sounds most like the 6L6, another tube I like a lot.  

That's another reason I like those QS.   Sometimes I'm looking for a different presentation and will re tube with a 6l6.    My last quad was 6l6,  was just missing the el34 tone a little