Analog problems require kickass solutions

Because it's an input signal not output, hence the line in

At a glannce, my guess is that the tape jacks are an input but the headphone jack is an output.

Even if it was an output, which it isn't as others have noted, headphone out and line out are not the same thing and can't be used interchangeably. Sound will of course come out of the headphone jack, but might have rather nasty side effects too.