Perhaps you are thinking specifically of ICBMs? There are other types of missiles, not all of which are rocket powered.

https://en.wikipedia.org/wiki/AGM-86_ALCM - "All variants of the AGM-86 missile are powered by a Williams F107 turbofan jet engine"

https://en.wikipedia.org/wiki/Tomahawk_(missile) - "After achieving flight, the missile's wings are unfolded for lift, the airscoop is exposed and the turbofan engine is employed for cruise flight."

https://en.wikipedia.org/wiki/Harpoon_(missile) - "Power plant: Teledyne CAE J402 turbojet, 660 lb (300 kg)-force (2.9 kN) thrust, and a solid-propellant booster for surface and submarine launches"

https://en.wikipedia.org/wiki/Kh-55 - "It is powered by a single 400 kgf Ukrainian-made, Motor Sich JSC R95-300 turbofan engine, with pop-out wings for cruising efficiency."

The hypersonic missiles use a rocket to get up to speed, like:

https://en.wikipedia.org/wiki/Zircon_(missile) - "A booster stage with solid-fuel engines accelerates it to supersonic speeds, after which a scramjet motor in the second stage accelerates it to hypersonic speeds."

Yes, since ramjets with performance such as this one were outside our engineering capabilities. As the article states, even this one was not possible a few years ago.

It’s also not entirely true, tomahawk cruise missiles for example use turbofan engines with a solid state booster.