Actually, the Variant: Familiars sidebar in the MM indicates that NPCs who can cast
find familiar might have a familiar, which could be a spirit like from the spell, or a crawling claw, imp, quasit, pseudodragon, sprite, or some other tiny creature. (That's all of the Pact of the Chain options plus crawling claw and "others".)
This doesn't say that the NPC uses
find familiar in connection with those creatures becoming it's familiar, but the fact that it references them having the spell at all really suggests that the spell has something to do with it.
Here's something from AL that might give a better idea of what they were thinking with all of these unclear statements:
That precedent allows you to cast a
find familiar spell on a willing existing creature (rather than summoning a spirit with the spell), and then the creature follows at least some of the rules under the spell--such as being only discorporated when reduced to 0 hp and able to be resummoned by casting the spell again.
I think that example should be used to make sense of the information in the MM--because it
can make consistent sense of it. So here's how I would run the rule to combine all this material together:
As an alternative to its normal usage, you can cast the
find familiar spell on any willing tiny creature that the DM approves. At minimum the list should include the specific creatures who have familiar sidebars in the MM, and the additional creature or two mentioned in the Variant: Familiars sidebar. That creature is now bound to you as a familiar, and, other than causing it to change shape into a different familiar, it follows all the familiar rules in the spell, such as being able to be resummoned if reduced to 0 hp, be sent to an extradimensional space, etc. If the creature has a sidebar about using it as a familiar, those additional aspects also apply (those creatures just make especially good familiars).
I can't actually think of any reason
not to interpret it like that, other than thinking it makes familiars too powerful or something. It makes the game more interesting and allows interesting concepts. And remember the downside that these special familiars have wills of their own and can terminate the bond at any time (I'd assume that should apply to any familiar that isn't just a summoned spirit). And finally, if you don't do something like that you have orphaned rules and precedents sitting there in the MM looking at you funny.
The only issue becomes how to keep Pact of the Chain familiars special. It's already pretty cool in that it is a spirit that can switch between those alternative familiar forms. But I would absolutely give it the extra abilities listed for any special familiars in the MM when in those forms. (That's contrary to Sage Advice, but that's a sacrifice I'm willing to make to make sense of the rules. And Sage Advice advises you to ignore Sage Advice if it doesn't work for you.) Personally, I'd also extend it to say it can take any other forms from the MM than that DM would allow to become familiars. That's all in the realm of interpretation.
As an outright house rule that contradicts the books, I'd say that all familiars (including the summoned spirits in animal forms right from the spell description) can attack normally. There is no good reason--it's not like their attacks are significant. For the Pact of the Chain, I'd say that when warlock gives them an attack, that attack (which uses their reaction according to the most up to date rules), is
in addition to their normal actions or attacks.
And that is how it's done.