there are at least two misconceptions here which render your discussion really pointless. first of all theres no particular reason for the band to have chosen this exact name. one theory says that bela couldnt find any bands beginning with the letter "ä" in music stores and that they randomly picked the name for that reason. 
 the other misconception upon which all your palaver is based is that there actually is a difference in meaning in both words, which simply isnt true for the german words. in german both doktor and arzt are used 100% synonymous and dont pay attention to academic degrees at all. arzt is the colloquial term for the profession, while doctor is used to speak to and about the practitioner as well. 
 also, arzt isnt as serious an expression as physician seems to be (at least from my understanding (native german speaker)). for example theres a saying that goes "feiern bis der arzt kommt", which means "(to) party till the ambulance arrives". dunno if theres an english version of this, but it can be used with any kind of occupation people would like to devote themselves to while forgetting their surroundings and without wasting a thought on problems that might arise after it. 
 id go with "the doctors" and not with the physicians. but not because of the english understanding of the word, but rather coming from the synonymity and casuality with which doktor and arzt are used in german. 
 of course doktor is also an academic title. but that is not important in this case. the emphasise depends on the context, and that is simply not given here (for it to be linked to the academic title). go with doctors. why? it sounds better. feeling for language cant be explained. but your nitpicking doesnt help here.