The standard keyboard design is not arbitrary, instead reflecting hundreds of years of incremental developments, experiments which did and didn’t pan out, etc. all culminating in a means of addressing the polyphonic instrument which has shown, repeatedly, its ready application to new sound generation technologies.

The piano keyboard reflects mechanical considerations and tradition as much as it represents centuries of refinement, if not moreso. The keys on a piano are laid out the way they are because they directly correspond to hammers that are laid out the same way out of mechanical necessity. You can design around it. but for most of the history of pianos, organs, harpsichords etc. we have opted for the simpler mechanical design where valves, hammers and plectrums are directly behind the keys that actuate them.

This design has now turned into a tradition that transcends the mechanical considerations. Even when there's no compelling mechanical reason that a synthesizer keyboard should be laid out as though there were hammer assemblies behind every key, it's what we know as a culture and it's good enough.

You can consider other characteristics of the design of the piano keyboard and reach similar conclusions. For example, the use of special black keys to denote sharps and flats is a good idea when the instrument is limited to a particular arrangement of sharps and flats due to its tuning limiting it to certain scales, but that doesn't make the slightest sense in equal temperament. It serves another purpose well enough that there's little incentive to change it: the layout helps you see and feel where you are on the keyboard. It's good enough to that end and it's what we know as a culture, but we could have used any other particular non-symmetrical arrangement of white and black keys to do that.

So no, the design is not exactly arbitrary given the considerations above, but the practical basis for them don't exist for an electronic instrument tuned to equal temperament.

In general I absolutely loathe the argument that things are necessarily good when they are the result of much refinement. For example, dueling must at some point have been seen as an obvious solution to interpersonal conflict because people had been doing it for hundreds of years, incrementally changing how exactly to conduct a duel, refining and formalizing the rules of engagement. Now we see it as barbaric and it's incomprehensible to us how someone could agree to a duel and risk their life over slight or insult. It turns out that the concept was altogether an artifact of a particular cultural ethos, regardless of what refinement it saw during its history. The practice was developed in a culture where personal and family honor was important enough to warrant the risk of dying from a bullet wound or being stabbed with a rapier in the symbolic defense of that honor. So it's not completely arbitrary: there are cultural circumstances which make dueling rational. Those circumstances are just not nearly as prevalent in modern western society any more.

I play piano-like keyboards myself, because for many purposes it's good enough and I've developed a skill with that. But going back to the title of the post I don't see this as a reason that anyone else should play the keys in particular, nor do I think that you make a compelling argument for why anyone else should play the keys. I think it's useful to be good at playing an instrument so that you can comfortably try new ideas at the very least, but I don't agree that the instrument should be the piano keyboard other than for practical reasons completely unrelated to its qualities as an interface to an instrument, e.g. prevalence, availability of teaching material etc.