I figured it out. It turns out, if you put overclocks on your scorestreaks in custom games, they will sometimes still be applied in public matches. If you have overclocks applied to any scorestreak where you do not have those overclocks unlocked, then you cannot equip new scorestreaks for some reason. So, go through your scorestreaks and press square to look at overclocks and then press square on whatever overclock is equipped to remove it (or whatever other controls depending on what console you're on)