What is the best approach to write a literature review paper?
Usually review papers are written by established researchers in that field. They already got good knowledge about the topic and a library to work with. In that case you won't read all the papers in detail again, just the parts you need. You might read newer/additional papers and yes, ofc you read the whole.

Writing a review while you are not familiar with the topic is common if you start working on that topic. Often profs want new students to write a review which means they have to read the whole literature and afterwards they are familiar with the topic. In this case you should read all the papers. In the end the prof usually gives additional input in case something is missing or wrong.

If you write the review "on your own" you definitely got to read every paper very carefully, otherwise you might miss important details. I advise against doing this.

Writing reviews correctly would be very important since it helps everyone who wants to get knowledge on that field. Sadly reviews are often completly full of errors.