"I was listening to the Big Fugue in G Minor demo, and it made me sad that it sounds so static."
(I have not listened to the Fugue in G Minor demo, but speaking generally:)
It needn't. Another difference between piano and organ is decay: piano notes decay and organ notes sustain indefinitely. One of the consequences of this is that the release of a note on an organ is just as apparent to the listener as the attack. Organists can take advantage of this by controlling the duration of each note -- we're talking about millisecond variations here. So if you have four 1/16th notes, you can do, say legato between 1 and 2 and the rest are detached -- giving the illusion of accentuating the first note. Or even if they're all detached from each other, making note 1 a few milliseconds longer than 2,3,4 will also give the illusion of accentuating the first note.
You can also make whole phrases 'airy-er' by increasing slightly the gaps/silences between notes. Or 'thicker' by decreasing the gaps. You can also make a phrase super-legato by slightly overlapping the notes (negative gap?) You can also apply this to whole passages. In the Great Fugue in Gm you could have smaller gaps at the strong beginning, make the gaps a little bigger in the more introspective passages in the middle to make them sound lighter, then make the gaps smaller again towards the end for a stronger sounding finish. You get the idea. Sure, it's not the same as velocity sensitivity, but there's also much more one can do to expressively phrase notes on the organ than one might think.
(An important factor is the size of the room and instrument -- the reverb of a large room will obscure these sort of subtleties. Although Bach played on large instruments all over what was more or less Germany at the time, the largest instrument on which he had a regular job was only TWO MANUALS -- in consequently less than gargantuan rooms. Now, huge organs are of course great fun, but smaller instruments do have a clarity and transparency that huge instruments may not. Tradeoffs yet again.)
And we haven't even gotten to manual or registration changes. I've heard some organists play a long piece like the Great Gm entirely on only one manual. WHY? Or don't even do a simple registration change like add a reed or mixture towards the end. WHY?
Some organists (and pianists and violinists...) play statically -- boring is not necessarily the fault of the instrument!
As an aside, one of Bach's "side jobs" was technically evaluating new organs (were they built well or not) -- he was quite expert in that technology. Think about it: a pipe organ in those days was easily the most complex machine humans were making at that time (certainly in the West). Arguably it remained the most complex machine humans were making until the latter 19th century. In other words, Bach was fiercely interested in the most advanced music technology of his day. Towards the end of his life he also had the opportunity to try out the newly invented pianoforte -- liked it very much but had some concrete suggestions on how to improve it (which the builder took to heart). Or, check out the Lautenwerck (http://www.baroquemusic.org/barluthp.html) -- Bach had one built to his specifications. In his own day Bach was a music technology freak!