Sound design is not EQing and mixing. That's EQing and mixing lol. However, it can involve mixing, if, for example, you are layering sounds to come up with a single "sound".
Sound design refers to the design of literally the most discrete unit of sound of a particular track. This is why most people generally refer to sound design as being able to program your own synth. It's note independent, meaning your sound design of a particular track USUALLY does not change note to note.
Sound design is prob 40% - 80% of what makes your music original. Structure, progression, eqing, compression, and mastering accounts for the rest. It is much more than being able to use a synth proficiently. You need to be able to use filters, compression, sweeps of all kind, modulation, wave manipulation, etc. literally anything at your disposal to get a sound that satisfies you. Using presets isnt sound design, its the first step in sound design as it usually narrows your choice down to a certain genre/sound your looking for.
Very very few people are masters at sound design. Many great artists know nothing about sound design, but they do know what sounds good. To master sound design is to be able to hear a sound in your mind and replicate it in reality. Its insanely hard to do, and might not be worth the time if that sound you hear in your mind sucks to begin with.
The truest method of getting a unique sound is as follows:
a). Get a midi keyboard or guitar (midi guitars r sweet btw)
b). Get massive, zebra2, synleth, electrax, nexus, etc. or if your a gear guy get an audio interface
c). Get vst saturators, tapes, Sausage fattener, camel phat, fabfilter volcano/timeless other effects/compression/eq vsts
d). Try everything under the sun. I usually like to set up my keyboard so midi goes to 2 or more tracks, load up some vsts, compression, and eq on all, then just play around. Put oscillators on some, try lfo panning, phasers, etc. Just keep trying and eventually you will find sounds that you like.
This is the way about 80-90% of producers find the sounds they like and eventually begin to create their own style.
Ok back to the question at hand: Is sound design important?
Yes and no. If your a rapper, you probably couldnt care less about sound design.
If your a producer into replicating certain pop top 40 hits, then probably not as well. You can get away with presets, the cutoff filter, ASDR, and EQ for most pop. Pop is, in my opinion, all about the hook, the melodic progression, and the how well its mixed and mastered. Sound design is secondary, though once again, many exceptions do exist. Anything produced by Dr. Luke, has immense amount of sound design. His ability to distinguish a pop sound and produce the cleanest, most easily listenable version is amazing. For example, the bass/pad in Roar by Katy Perry is a sound design masterpiece, and has either 4 or 5 layers including a sine wave bass, bass guitar, and pads all layered and eqed in a way that makes it all sound like a single compact bass.
If your into Arca, and post modern hip-hop electronica, then sound design is where it's all at. Most of Arca's music is composed of 5-6 sounds that filtered and animated in a way I can't even begin to describe. If you're trying to make a unique sound, you should definitely check him out. (on a random side note, he's 24 doesnt promote himself much besides coming out with an EP every 2 years, but he was a co-producer on Yeezus, and that speaks for itself).
To someone like Kanye, sound design is a part, but not the whole. To him it's a means of distinguishing one's music, but he probably spends more time on composition and editing (Stronger apparently had over 20 different versions, though not nearly that many different leads).
Another good example of sound design is Skrillex. Skrillex's main distinguishing factor was the sound design of his vocals, and his frenetic, all over the place, composition. The actual sound design of his bass and leads, however, was bland. He probably spent a lot more time on composition rather than sound design.
Bassnectar is almost the opposite. He is all about sound design though his compositions are pretty straightforward, as are his chord progressions and sequences.