Inspiration as part of the core? Wow, bold move there. Given that this mechanic is a 100% new thing for D&D, I would have expected them to put it in a module.
Once again... this comes down to the question of "What is 'core'?" What does 'core' mean? Does it mean it's a rules system that the game
cannot function without (like ability scores for example where you can't just remove them in their entirely and have the game still work)... or does 'core' just mean rules systems that will be listed as part of the main body of the game and how it's presented, but don't have to actually be used if you don't want to.
Backgrounds will be in the 'core game' it sounds like. When the book presents the list to go through to create your character... Backgrounds will be included in that list, and its chapter will probably fall right after the Race and Class chapters in the book. But if a DM chooses NOT to use Backgrounds... can the game still function? From the sounds of things... sure. It's not a "module" per se, because "modules" seem to be options that either modify/change the basic rule system (and will probably be sidebars to those rules systems in the book), or they will be complete add-ons that will show up in different parts of the book (or different books altogether) and which most tables are expected to probably not use.
So Backgrounds seem to be part of the "core of the game"... but the rules are not
required to use for the game to still function. Inspiration sounds as though it is the same sort of thing. It's not going to be a small rule system or modification buried within the DM's Guide that a DM can choose to add if he wants... it's going to have a place in the line-up of chapters that WotC believes to be in the typical D&D game (if we have to, we'll use the dreaded term "default" here.)
Your standard, default D&D game if you follow the character creation design parameters down the line with no changes will have players determining bonds, ideals, flaws, and the key problem/question of their character (in addition to rolling for ability scores, and then choosing race, class, sub-class, background, and equipment). But if a DM doesn't want to use those things... he can probably pluck them out without any problem and it won't impact how the game works on a mechanical level whatsoever.
It's not "core" as defined as a primary game function that is required for the mechanics to work on the whole... it's "core" as defined as one of the subsystems that will be listed as part of the default series of systems that your unmodified game will have you use. But which is easily removeable if you so choose.