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

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.  

I have an EL34 headphone amp that sounds quite good.  Among pentode/tetrode tubes, my favorites are very low powered tubes.  I run 349 output tubes in pushpull that deliver about 4-5 glorious watts per channel.  This is my favorites tube aside from the truly crazy rare 252 tube.  My next favorite is the 350B (a variant of the 6L6 tube), followed by the 6L6 or KT66, and then EL84 (a tube often used in less expensive amps, but, I’ve never heard an EL84 amp I didn’t like).  
 

Of triode tubes, either pushpull or single ended, I particularly like the 45 and 2a3.  

Inna,

My speakers are S.A.P. J2001 (twin 12” Alnico woofers, bullet tweeter and midrange horn sitting on top) but the midrange horn and compression driver replaced with a Western Electric 713b driver and KS 12025 horn.  The system is 99 db/w efficient.  

I suppose everything is important.  There are drop-in replacements for the 349 that sound very good, and I could easily live with the MUCH cheaper alternative.  However, the 349, as used in my amp, is run very gently and will last a very long time so I don't worry about using it up I have spares).  The input/driver tube for my amp is the 348 tube which is also a very expensive tube.  It too is better than its drop in replacements, but that drop in replacement is very close in sound and is extremely low in cost.  It is almost impossible to find 348s being sold.  The last pair of supposedly NOS tubes I saw sold for $2,000 each (I need four), and some crazy person was selling dead 348s to Western Electric collectors for $1,600 each.  The drop in replacement sells for around $8 a tube.  These days I run the $8 tube in my amp and in my linestage which also uses the 348.  My linestage also runs 310a and 310b tubes, and these are becoming pricey too.  

How much one is wiling to pay to run the very best tubes for a particular amp is a matter of personal priorities and finances.  I know people with magnificent clones or rebuilds of Western Electric 124 amps that run 6L6 tubes in the amp because the better sounding 350B tube is too expensive, in their judgment.  I can understand that choice as the 6L6 in these amps do sound pretty good.  

For my personal taste, I will take a 124 amp (pushpull, two output tubes per channel) running either 6L6 or 350B over all of the 300B SET and parallel SET amps I've heard.  My 349 amp is a clone of a Western Electric 133 amp and sounds very similar to a 124 amp.  It has the same output transformers as 124 amps, but it also has input transformers that the 124 does not have, and it outputs a little bit less power.