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

Agree with most posters here, EL34 easily my favorite push pull amp tube. Sweet, natural mids, highs may or may not be rolled, depends on amp, bass is biggest issue. Bass issue is lack of slam, can be too resonant in some situations. Power supply in amp/speaker choice key here, get this combo right and bass can be ok. Still any KT will beat it on slam, extension, problem with KT is never get that same sweetness, natural mids, highs can be pretty nice, mids are the issue for me with KT.  I also generally agree the lower power KT sweeter than higher power.

In the end, the EL34 and push pull didn't do it for me, transitioned to 845SET than 300B SET. 300B is a magical tube with the right speakers and properly engineered power supply in amp, with this no roll off in highs, no soft, overly resonant bass, some totally misguided on stereotyping 300B sound. The boutique 300B tubes absolutely necessary. Running 300B in parallel can be replacement for those running EL34 with speakers SET can't handle, some very nice amps in this category out there.

If midrange is not right, the rest doesn't matter.

Power cord can make quite a difference for dynamics and bass, by the way. I use older Purist Audio Dominus Ferox, appears to be a perfect match.

Absolutely correct about 300B amps. So much is the power supply. In fact, it took an expensive, properly engineered 300B amp to open my ears to this tube as viable. Have never heard a good 300B amp for less than 15K. 

in my experience:

in General, Absolutely.

The VAC 80/80 was KT88s, good sounding $2,500 amp but it chewed up KT88s, and it bias drifted. I had an MV52, that was sweet, perhaps beige like the front panel, but lacked any real punch.

I also used a VAC Ren 30/30, gorgeous but it chewed up 300Bs and a Ren 70/70 - outstanding in every way but broke the bank chewing up 300Bs - 8 at a time!

I still have the 30/30 (no tubes), it has something awry as it makes very unpleasant noises, probably a cap is blown!  I met Kevin Hayes at Sound by Singer (early 2000s), he opined that the 30/30 might have been the best sounding amp he had ever designed, but it was not a commercial success, he could not sell it for a price that was profitable.