Your strategy sounds sound to me. You cannot read the papers from end to end. You need to weed out they papers you will possibly need in your review with as little effort as possible. Title and abstract should be enough. Once you have done this you should be left with a bunch of papers that are likely to be of use to you. While you have looked through them you must also consider how the material can be organized. There are many ways of doing this, chronological order (boring), thematic order (are there subfields?), contradictions (are there compeeting ideas) etc. The point is to try to get the information organized. You need some form of organization fo the papers so that you know how you will treat them in your review.
Once you have found all papers you think you need you can start assembling the information into your text. It is at this point you can start reading them in more detail. It is still not necessary to read every detail. you need to focus on the background to the different conclusions drawn.
The goal of the review should be to provide a new perspective on the field you have chosen so your own familiarity with the field (papers) is key. Try to collect as many as you can; not all will be used.