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My silly piece of advice isn’t about your question of “all at the same time– but slowly”…. versus…. “one at a time– mastering the one you’re working on”— (if I understood what you were asking)’
My opinion is more about one of the steps– regardless of which way you do it.
1. Monkey around with the song- (I mean REALLY mess with it!)- AS you are practicing it. Change a chord/ change the key/ changes the strumming pattern…… etc, etc. You will really learn a song IF you mess with it some. Your brain is gathering ALL that info as you go– even if you don’t use anything you changed or messed with.
2. Learning and Practice. They are two different things. Recognize the difference between the two.
3. One trick I use: Most people start practicing/playing a song at the beginning– but then get hung up somewhere; they then go back and start again…. back at the start. (so…. the end of the song gets waaaay less reps than the beginning). Practice the song from the back to the front— this way the end-part of the song gets the same amount of— and you are just as confident about the END of the song– as the beginning.
Anyway– just my two cents