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

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

I think it’s hard to make an EL34 not sound sweet. You have to really mess it up. The larger KT tubes will (by comparison) give you brute force, dynamics, impact - but it’s hard to keep all that while getting them to sound as "sweet" at the same time, as the smaller tubes which seem to do it so naturally (EL34, EL84, 6L6’s etc). I think VAC’s recent high-end lines do a GREAT job at that (plus next-level articulation and refinement), but takes some doing and ain’t cheap.

Vintage tubes amps are a lot of fun and a nice break from the "hifi" sounding amps because they’re so inherently musical. Part of that, I think, is the use of those tube types. But there are those limitations - power, and usually bass impact / dynamics.

Matching and preference will play a role which one you like.