Ripple current Buck converter
You've answered the question yourself - "with given values".

The amount of ripple current is defined by the buck inductor value and the volt-seconds applied to it. If you change operating point (different input voltage, different output voltage, different switching frequency) or change the inductance value, the ripple current will change.

Why is there so much ripple? That's how the circuit was designed. A DCM buck would have even higher ripple, which is a trade-off for simpler loop stabilization. If you want less ripple, make it CCM and design your slightly more complex loop accordingly.

The amount of current ripple applied to Cout is what defines the amount of ripple voltage you get on the output. So yes, in a manner of speaking, Cout "takes care of" the ripple current in so far as it deals with whatever's applied. If Cout is too small, your output won't stay in regulation.

You haven't specified enough about your application, but there is no reason why a 300mA buck operating at 247mA of ripple current couldn't be stable. It would be barely in DCM mode, but that's fine as long as your output capacitor ESR could handle the ripple and your application is OK with the ripple voltage.