You are right. This is one of the (many) cases you may click on "Report a problem" and "My answer should be accepted". I always use uncommon translations to test how experienced DL is. Some kind of challenge on the way ..
Ich will --> in the sense of want should call for the dative preposition, but why is it that in this case, when it implies going somewhere it does not call for the accusative preposition? " Ich will nicht zu den Arzt" is incorrect.... but I just wanted to understand why.
Two problems. First, it should be gehen, not gehe. Writing ich will gehe is like writing I want go. Second, the participle gehen should go at the end of the sentence. Only the conjugated part of the verb gets to sit up front (in 2nd position).