Everyone is different. Players anchor their picking hand in several different ways - most popularly either at the bridge as your instructor is suggesting, or by hooking a finger around the bottom of a pickup as a reference point.

I think you're talking about Chris on the red strat... and it sounds like he goes the same way as me - no anchoring at all which means your hand can move freely across the area to get different tones. It's also useful for some of the more advanced techniques when you get there.

Like so many other aspects of playing guitar, there is not so much a 'correct' way as a personal preference. If it works for you it's right. From what you're saying on muting etc, it sounds like you need to become more familiar with it to get the muting down.

You may find anchoring your picking hand a limitation further down the line - all I can really suggest is have an experiment with it and see what feels most natural for you. You should be able to find the string you need in time whether you anchor or not.