It depends a bit on the definition of a "wing tip vortex".
If this is defined as just the local flow around the wing tip, it is too narrow to encompass all 3D effects and you are right.
If, however, the term "wing tip vortex" describes the beginning of the whole wake rollup process caused by downwash and sideways flow, then this does cause induced drag. The vorticity behind the wing cannot create drag anymore; this can only be done by things happening directly at the wing. But induced drag is spread over the whole wing and not restricted to its tips, so the "wing tip" part of the name is misleading.
In my view of the process, induced drag is caused by the gradual bending of the flow by the wing. By bending an already deflected flow further downwards, the reaction force must have a backwards component, which is induced drag. Note: No wing tip flow required for the explanation! But that is too simple, or pure 2D drag would also produce induced drag. In fact, the flow is first bent upwards ahead of the wing (induced angle of attack) so the bending happens not beginning with straight but upwards moving flow. Without the added downwash of the trailing vortices, all effects cancel out so only a vertical force is left. Add the downwash over the wing produced by the free vortices and - bingo! - the backward component is precisely the induced drag added in 3D flow.
The often-heard explanation that the flow around the wingtips causes induced drag is much less helpful and similar to saying that wet streets cause rain. In the end, wingtip flow is only a reaction to local pressure differences, in the same way that the induced angle of attack is. And it is this pressure difference over the whole span which causes the flow to bend around the wing and be accelerated downwards. This downward movement then causes wake rollup and the wake vortices, which in turn start at the wingtip.
As to the "why" you are asking: In my opinion it is copying from others without fully understanding cause and effect.