Really, the wizard's entire benefits are basically in their spellbook. A spellbook that any character can get at level 1.
Not really -- a number of the spell school abilities are also really good, such as the divination wizard's Portent ability, gained at level 2. If your Portent dice roll high, you can use them on spell attack rolls or concentration checks, and if they roll low (which I think is better), you can use them for enemy saving throws versus your spells.
Some other useful abilities are the conjuration wizard's class ability not to need to roll concentration checks on damage when concentrating on conjuration school spells, or the transmutation wizard's Philosopher's Stone. YMMV, of course, but yeah, I definitely would not say that "the wizard's entire benefits are basically in their spell book".
Another point, however, is that sorcerers are full casters and dipping into the wizards do not interfere with their spell slot acquisition rate. The only thing that is reduced is sorcerer points and the higher level spell known that you would've gotten.
A sorcerer doesn't gain any new spells known when advancing as a wizard, though does get a spell book with a number of 1st level spells, as otherwise explained elsewhere in this thread. Up to you as to whether a bunch of 1st level spells is better than the higher-level spell you would have gotten going up a level of sorcerer.
However, taking this dip at, say, level 7 lets you put all your spells known into scrolls which can then be put into your spellbook. This lets you "forget" the spells on level up without actually forgetting them, they're now wizard spells.
This is more limited than you believe it is -- sorcerers don't prepare spells, so when preparing spells, you use the wizard rules for preparing spells, and you prepare them as a wizard of your wizard level, not as a wizard of your character level. For starters, this means a Sor7/Wiz1 only prepares a number of spells equal to that character's Int bonus plus one. So even if the spell book shenanigans work the way you describe, a 16 Int multi-classed wizard could prepare just 4 spells from that book, which is a far cry from 'all of your spells known'. And unless you're using a system other than point-buy for your starting stats, I have a hard time believing that your primary sorcerer has better than a 16 Int at level 8, unless you're deliberately gimping yourself as a sorcerer to do so (or were lucky enough to find a Headband of Intellect, but even that only gives you one more spell).
Also, this isn't Third Edition any more, and 5E wizards don't get Scribe Scroll as a bonus feat -- if your DM allows you to create spell scrolls as a downtime activity, you can do this even if you're not a wizard. Note as well that the rules for copying a spell scroll into a spell book require you to make the DC 10 + spell level Intelligence (Arcana) check every time you want to copy a scroll (DMG, pp. 200-201), and the scroll is consumed regardless of whether you succeed or fail at the check. (This is one of those situations where the divination wizard's Portent ability would actually come in handy...)
A sorcerer also gets the ability to add spells to their overall list via spellbooks and scrolls external to their character. That way, if a sorcerer feels like they don't have enough, the DM is free to give them more.
That's not exactly how it works, either -- having a spell in a wizard spell book doesn't make the spell a sorcerer class spell, but that's not really necessary either, since the character is also a wizard and thus has all the wizard spells as class spells by default. (Yes, even spells that aren't in the wizard's spell book are still wizard class spells if they are on the wizard class spell list.)
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Pauper
