If you go to music school to learn violin, saxophone etc. you will have mandatory piano class, but pianist won't have mandatory violin class.
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This is mostly because solo violin does not teach enough coordination. Though concert performance practice may help with that it is still less involved. Performing sequentially compared to parallel.
Accordion is closer but still rarely uses polyphony.
I don't think this has anything to do with coordination. Piano is considered the master instrument. You can pick something which require more coordination and you still need to take piano lessons.