"That would be great if that dt conversion was automatic...but it's probably utopic!"
well, it would be hard to know what needs dt and what needs NOT dt (if you move once an object of 200 pixels, dt is not something you might want).
just have an event, compare situation A, and Situation B (the difference being a time between the two), for the same time, you should arrive at the same situation B in both cases in you calculs regardless of the framerate.
btw, just for the "I said it" aspect of that (I know I am boring with that, but now that I can prove it once and for all, I won't let that chance go):
A=lerp(A,B, 120*dt), for 1/30 seconds (2 frames at 60 fps, 4 frames at 120 fps)
at 60 fps, A is first equal to 0, in one frame: A=lerp (0,B, 2)=2B, in two frames A=(2B, B, 2) =0 (if we continued, we would go 2B to 0 to 2B to 0, etc..)
at 120 fps, A is first equal to 0, in one frame: A=lerp(0, B, 1)=B, in two frames, A=lerp(B, B, 1)= B+1*(B-B)=B
in 3 frames, A=lerp(B, B, 1)= B, in 4 frames, A=lerp(B, B, 1)=B (if we continued, it would still be B).
not the same result, so framerate independancy is not there, in short, never use dt in lerp like that, never use lerp like that for that matter, as it is not intended to be used like that!