The DMG says the following (p. 241)
Inspiration might not work for your campaign. Some DMs feel it adds a layer of metagame thinking, and others feel that heroism, roleplaying, and other parts of the game are their own rewards that don't need incentives like inspiration. If you choose to ignore inspiration, you're telling the players that your campaign is one where you let the dice fall where they may. It's a good option for gritty campaigns or ones where the DM focuses on playing an impartial role as a rules arbiter.
Also, as you already mentioned, the PHB sees Inspiration as something that can be used. The DMG also uses these words
Think of inspiration as a spice that you can use to enhance your campaign
While Inspiration is not described as an optional or variant rule, it is completely valid for your DM to ignore it.
Regarding the question how much harder you make your game, I would recommend you to see Inspiration from a different perspective:
Inspiration is a Reward. Something you get for taking a risk or doing something great.
So to get inspiration you are expected to do something. You do not get it for free. Your DM could encourage risky play with it to balance out the risks you take for gaining Inspiration. If he doesn't, just don't take the risks, so they don't need to be balanced out by Inspiration.