[Disclaimer: As this thread is about DM praxis, everything I'm saying here is just the way I like to run things. It is based on my personal experiences and preferences, and is not meant to be universal GM advice. Also, all rules have exceptions (including this one). Also I have all the spell cards.]
I've been playing these games for 30+ years now, and I have never met anyone who could learn all the rules that apply to any character more complicated than, maybe, a 1E fighter. Heck, I don't think I've ever actually seen anyone learn all of those, if you include speed factors and the special cases in the number of attacks rules. I mean, sure, you can be basically familiar with them. But you're not going to be able to remember, with perfect reliability, all the edge cases and questions about phrasing that might apply. Especially not for spellcasters.
I don't care about the exact wording of the rules. I'm not running Jeremy Crawford's game; I'm running
my game. The rules belong to the DM, and are just a small part of his toolbox. Whether we play "correctly" by the book is not important to me. The important thing is that there's a type V demon trying to mince you, and if you hesitate, you die.
So, okay, "write it down", but if we're writing it down, why not just use the one that was already written down, which is to say, the book?
The book has lots of dumb rules and content that isn't relevant for your character. It is inefficient to look everything up in a 300 page book when all you need is 1 page worth of information. If you can take 20 minutes of your own time to put together that page, it'll save us all a lot of time when we're actually playing the game. This is a pretty basic requirement of the game as far as I'm concerned. This is what the character sheet is for.
Edit: To clarify, there are three types of rules in the PHB. As a player, you need to have your character-specific "content" rules (
e.g. race and class features, spells) available for immediate reference, and you should understand the "how to play" rules (
e.g. saving throws, bonus actions, concentration). The players don't need to have a deep understanding of the "how to resolve actions" rules (
e.g. jumping, grappling, speed factor, surprise); that's the DM's job.
but if I'm playing a caster, and someone asks a question about the boundary cases of one of my spells, I'm gonna look it up. I am not going to have 100% confidence that I remember the exact wording of how teleport familiarity is defined, or the exact restrictions imposed by a grapple, or whatever else.
If someone has a question, they ask the DM. The DM answers the question using any of the many tools at his disposal. Depending on the needs of the moment, that answer can be anything from a snap decision ("you can't do that, you're grappled"), to a die roll ("50% chance you're in the fireball"), to a rules reference ("teleport is intense, look it up"). The player does not need to open a rulebook unless the DM decides it's worth taking the time.
So.... You're good with me flipping though essentially a hand written book, but not the real book.
Short answer: Yes.
Long answer: If you feel you have to write out a whole book just to know how to play your character, maybe we're not on the same page. We
are talking about D&D 5e here, right? Even the official character sheet with all the fixins is only 3 pages. Again, if you can't handle this, I have plenty of pregens with all the information you need.