My 5E player's Handbook (2014 issue) reads a little differently than the previous quote. It says in the "Spellbook" sidebar, page 114:
"Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it."
So here it is based on slots, not preparation, and the multi-class slot rules are pretty clear. What I like most about the Wizard 1 multi is that you have the ability to learn from a scroll. That is part of the fun... to find a scroll you can't use yet, then eventually get to a level where you can learn and cast it.
I agree it sounds too good to be true and I didn't believe it myself until I tried it in BG3, but on the other hand, with unlimited direct casting from scrolls by any class, cleric domains, druid circles, and Bard magical secrets, I guess it kind of fits in with the 5E modus vivendi.
