You are right, it is fine to use "on the other hand" without explicitly mentioning "the one hand" - the reader/listener can easily infer it.
Indeed, using "on the one hand... on the other hand" in most contexts sounds laboured and overwrought. It is sometimes useful to signpost to the audience in advance that you are going to supply an opposing view later, in which case it is useful - but for the most part it is better to leave the first part out.
In support of this: I spent a little time perusing the British National Corpus. It reports 5311 uses of "on the other hand", but only 1417 of "on the one hand". That would seem to suggest that "on the one hand" is only used roughly one third of the time, but in fact it's even less than that, because (judging from a random sample) most uses of "on the one hand" contrast it with "on the other" (or not at all) - so it's probably closer to one in four.
As this NGram shows, thumb was once less common than fingers. For some reason people have never been particularly keen on thumbs or finger, but they get used too. As Neil Coffey commented, fingers remains more common in British English.
Green hands do in fact exist too, but sometimes simply alluding to the inexperience of general-purpose unskilled employees. I think thumb wins out in the end because the singular is a bit simpler, and thumbs don't have so many other associations that might confuse the imagery.
My chart combines US and UK usage figures, but looking at the patterns for each country separately makes it clear that US thumb and UK fingers became far more dominant in their respective countries from about the 1940s, with little change in usage figures for whatever word was being used on the other side of the pond. I think this may be a result of increased commercialisation of "domestic horticulture" at the time. This probably led to a proliferation of books, magazines, local clubs, etc., and these by their very nature would be far more likely to encourage standardisation around an easily-recognised term.
Thus it may be little more than a fluke that the US standardised on a different term - if neither was particularly well known before, either would be happily embraced by commercial and social networking interests. But standardisation would be very much encouraged thereafter, whichever form happened to have taken the early lead.
Obviously a gardener spends a lot of time handling plant material, much of which is green and might therefore stain one's digits. Since the metaphorical usage is so transparent, it could be re-coined many times. It would normally be understood on first hearing, and often passed on by the hearer, so the expression could have had multiple opportunities to both germinate and thrive.
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So far from my research, this seems to be the basic idea; that a person being "waited on hand and foot" has handmaids and footmen (or their equivalents) to perform any manual labor the person would otherwise have to do on their own.
The term may have different but related origins; it could be related to a similar term viewed from the other side, that a person is at someone else's "beck and call", responding immediately to any gesture by the person being served. An extremely attentive servant or corps of same could respond not only to obvious hand gestures, but by more subtle movements of the feet.
Lastly, it's perfectly valid to think of it in the more modern sense of being pampered physically. To "wait on" someone or something is to be immediately available to answer any need. The term may thus have originally been "to wait on someone's hand and foot", thus meaning to have no other duty but to address any need of that hand and/or foot, be it heat, cold, an itch, or in more general terms responding to its every move including as a gesture having meaning. Over time the possessive may have been discarded.