To attempt to make further analogy, and suggest that 'if bowing is left, surely the dojo, the gi, the hakama, the terminology etc must be left too', is illogical. Why? Because these other elements are purely Japanese, there is no shared culture which could lead to problems of this nature. The bow is not particular to Japan; it has Judeo-Christian-Islamic connotations and significance, and therefore there must be allowances made for people of those traditions to adhere to their cultures' interpretation of it.
Let me put it this way. A keiko-gi is, in the end, a training suit. Functionally, it's not different from a sweatsuit. All other cultures have such suits, of various fabrics and weaves, but of the same function. Hakama are simply pants, which we have in western cultures as well. The terminology are simply words. Very simple, very pragmatic descriptions of the technique. To a Japanese person, "Shomenuchi irimi-nage" simply means "Frontal strike entering throw".
Likewise, the bowing is Japanese as well. A keiko-gi and hakama is a choice. One chooses to wear Japanese style clothing instead of other cultural clothing that would suffice just as well. The terminology is a choice. One chooses to say the words in Japanese rather than perfectly acceptable translations that would actually impart the meaning even clearer than the Japanese does. And likewise the bowing is choice. One chooses to bow in the Japanese manner (like using Japanese clothing and Japanese words), or one chooses to bow in one's own cultural manner.
At heart, I imagine this is a fundamental difference of conception. In my opinion, the physical act is essentially empty. One does the same movements in order to look under a couch. What makes the bow meaningful in any way is what mental and spiritual intention the bower puts into it. If you're thinking about the electric bill when you bow, it's not a proper bow to God, even if you do it in a church, synagogue or mosque. Mental and spiritual intention make a bow to God different from looking under the couch, different from tripping and falling prostrate on one's knees, different from a back stretch, and different from a bow in the dojo of a Japanese art, wearing Japanese clothes, and using Japanese words.
In both western culture and Japanese culture there is a gesture wherein a person, holds their hand out to another person, palm down, fingers slightly curled, and then repeatedly moved back and forth. In American and English culture, this means, "Shoo, go away." In Japanese culture it means "Come here." When and where does it mean which? Whichever context it's done in.
In both western culture and Japanese culture, there is a gesture where a person waves their hand back and forth in front of their nose. In America, this means "Yuck, something stinks." In Japan, it means "No." When and where does it mean which? Context.
In both western and Japanese culture, there is a gesture where a person spreads their fingers out, and then makes a circle with the forefinger and the thumb. In America this means "OK". In South America, it's a vulgar insult. In Japan, it represents money. When and where does it mean which? Context.