twocats I have not read all other answers that you have got, but if you are interested in finding out for yourself if your friend is correct, I would test it. Start with a short piece of your level - or maybe even memorise only half of the piece, and find out for yourself.
For a very short time I had a teacher who demanded memorisation, so I did try it. For me, it adds absolutely nothing. The piece did not become a part of me. When recording the piece, there are already so many things to think about, and adding to that the stress that I would not remember correctly, did not do my performance any good. However, maybe I should have spent even more time on that piece... (which would make me progress even slower...)
About your comparison: Always having the score is like driving home by way of a very complicated route, and knowing that you can glance at your GPS whenever you need to.
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... feeling like the pianist on the Titanic ...