Thinking about 'bleeding red' all over a student's paper during a tutoring session makes me cringe. I think there are many other more effective strategies for fixing mistakes and making corrections. Having the student read the paper out loud struck me as one of the best ways to start a session. As the Longman Guide noted, this immediately gets the student involved in the session so they are not sitting there staring numbly at the wall while you read (and use the dreaded red pen to slash away). Rather, when the student reads out loud, it gives them a chance to hear the mistakes for themselves: where the verb tenses may not agree, where a preposition doesn't work, where a sentence just keeps going...and going....and going - and if they can hear the mistake, they can usually fix it themselves without even needing you to note the correction.
Another thing that I think is helpful to do while the student reads out loud is to follow along with them, pencil or blue-or-black inkpen in hand, and simply underline passages that might be confusing, unclear, or misplaced. That way, you'll have an idea of places that might need a second look; you can direct the student's attention to those areas and ask them leading questions like "Hey, as your listener, this part was a little bit unclear. What point did you want to make with this?" or "This is a great idea, but it doesn't seem to relate to the rest of the ideas in this paragraph. Do you see a spot where this might fit better, or do you think this might need its own paragraph?"
As far as meeting the student's needs while still staying true to the idea of a tutoring session - not an editing session - I think the best thing to do is set out expectations and guidelines early! If we can find out what the student is looking to do, then lay out a basic plan for how we can meet their needs and help them with their assignment in a way that doesn't involve adding and subtracting commas, then we've set the session up for success. Everyone has a sense of direction, so no one feels like they're just floundering around working on random things as the tutor pulls ideas out of thin air; the session has a point and a goal.
However, once the goal of the session has been established, I think the tutor "influence" needs to be pulled back a bit - once the student has an idea of where the session is going, it can then take the form of a conversation between writers instead of a teacher-student relationship. That way, the student feels like they can take ownership of their text instead of having the tutor own it for them.