That’s a complex question. In general, I would agree with Chysn and say that one can of course write new pieces on all kind of instruments. However, cultural contexts of the past are significant when it comes to how we perceive particular instruments, or more precisely: what we tend to associate with certain instruments. Hence, historical contexts and pop-cultural references continue to play a big role.
With regard to the synthesizer in particular, the question about old vs. new also touches upon the infamous discourse about whether a synthesizer should have a keyboard or not, as it has been exemplified by the case “Buchla vs. Moog” (as this is generally too bold, I put it into quotation marks). No matter on which side you are, no matter what your musical and performative preferences are, I think that it can be agreed upon that Buchla had a valid point insofar as he was coming from a historically justified perspective when problematizing the attachment of a keyboard to a synth, since all instruments of the past always used to have their own and unique ways of being played. So I think, at least in an objective manner, him questioning why the synthesizer of all instruments should be associated with the keyboard goes back to a reasonable objection that is not exclusively tied to any kind of avant-garde agenda, but to a deeply historical consciousness about the peculiarity of musical instruments.
The same goes of course for keyboards themselves, as a keyboard is not always just a keyboard. In the case of the piano, the keyboard is the expressive source of how a sound is being triggered and articulated in the most fundamental way. In the case of the organ and most keyboard synthesizers, the keyboard is a simple on and off (or gate) switch that doesn’t allow for any sonic variety and certainly does not set in motion any sonic waves. Even Aftertouch and velocity are just very simplified workarounds that don’t reach the complexity of the manner the keyboard acts on a piano, except one has the option to assign dozens of modulations to those input sources that can approximate the complex and subtle variances the piano keyboard is known to control.
Of course I took quite a detour from the original topic, but I just wanted to refer to the underlying and general problems that are somewhat related to your more specific question.
Aside from that, I think that the question about how easy or hard it’s going to be to make new or different music with all the instruments that have been named so far, depends very much on the preconceived notions that are attached to those instruments. I’d say it varies in each case.
The pipe organ, the first mechanical additive synthesizer, is quite an extreme case, as it’s mostly not only associated with a certain type of genre (sacred music), but also the according place (church). The majority of people probably don’t think of composers like Gustav Mahler, who used the organ in conjunction with his massive orchestra, or György Ligeti (the piece “Volumina” has to be mentioned as one of the most avant-garde pieces for a pipe organ here I guess). To a slightly different but yet comparable degree, the same goes for the harpsichord, which is almost exclusively associated with the Baroque era although its sound has certainly made it into some pop classics, most notably “Golden Brown” by the Stranglers.
The acoustic piano on the other hand has been pretty much continuously present in all kinds of contexts from the days of its introduction. You’ll find it in classical, romantic, and avant-garde pieces as much as in all kinds of popular styles like Jazz, Pop, Rock, Alternative, and whatsoever. Part of its popularity is also related to its career in bourgeois households from the 19th century onwards and the fact that it was regarded as the one-man orchestra substitute due to its wide tonal range and playability (the tradition of the piano arrangement of almost any orchestral piece is a testament to that, which has not only popular or commercial reasons, but practical ones in the daily rehearsal life of any opera house).
The Hammond organ is again more specifically associated with particular styles. These still cover genres like Gospel, Jazz, Rock (both mainstream and so-called progressive), and Pop. But it almost continuously evokes a familiar notion just like the famous trademark sounds of a Gibson, Gretsch, or Fender guitar.
Regarding the connected associations, the Mellotron is probably amongst those instruments that are most determined by the historical context within they first appeared in. Being one of the first sample playback devices, its sound is characterized by the rather unsophisticated technology of the day plus I’d say that it always already sounded old from the get-go. It just has that ‘playing an old record’ flavor to it. And yeah, all those trademark sounds, like the flutes, strings, and choirs makes you either immediately think of the Beatles’ “Strawberry Fields Forever” or various Prog hymns from King Crimson to Genesis and so on.
As for the Minimoog I’d say it depends on its use. Being a synthesizer, it of course has the advantage of being adjusted in different ways. It doesn’t always have to appear as the Keith Emmerson solo machine. In fact, there are lots of Brian Eno records on which you wouldn’t even recognize it. And rather current bands like Portishead, for example, also don’t make it sound like the typical Prog solo machine. So time and many recordings have proven that it can also be used as an ensemble machine (that integrates well with a band’s sound instead of being at the forefront) and with different sounds and timbres that don’t scream: “I’m the famous Minimoog that does those funky basses or Prog leads.”
The Arp Pro Soloist is a different matter, as it’s rather a preset machine with – for its time – flexible real-time tone articulation controls (mostly related to its Aftertouch function). Other than that it’s not really a synthesizer, as it doesn’t provide the user with the tools to synthesize anything. That it is mostly associated with Tony Banks has only to do with the fact that he was amongst the very few rather known players who used one due to not wanting to replicate the Minimoog standard that was already established by then and I guess also due to the rather practical fact that he didn’t earn much money at around that time. I think David Bowie used one on “Low” or “Heroes” too, but not as recognizably. Banks’ choice in that regard says also a lot about him not really being a synthesist, but mostly a keyboard player that occasionally processed sounds (mostly his Clavinet and RMI electric piano with a fuzz box and his organ with a Phaser) and who later naturally welcomed samplers as the perfect solution for him. After all, he’s not really known for having had a great affinity for programming. He might have used a whole lot of classics later, but mostly with presets. He even had the studios’ ARP 2500 and 2600 being programmed by producer David Hentschel. Even on the Prophet-5 and 10 he mostly used presets or slight variations thereof. I think his most notable piece as a synthesist must be the piece “Who Dunnit?”, where he played the Prophet-5 while switching patches, which produced funny noisy artifacts. Dave btw. always wrongly attributed this to Brian Eno in the past.
So while I would generally agree with Chysn, I do still think that the question about how easy it is to produce new or different music on these instruments varies with regard to each instrument’s history and its (pop-)cultural connotation. Of all the named ones, I think that the piano and the Minimoog probably allow for easier solutions across different genres while it’s more of a challenge regarding the other ones.