It takes a single Sunburn [0] to take out a carrier. There is no defence - yet. Of course, when the railgun and lasers are deployed, this all changes again, but in the meantime the US' carrier fleet is principally a policing tool, and would be dead in the water during an all-out war - assuming the US chooses to go against an enemy that has the Sunburn in its arsenal.

[0] - https://en.wikipedia.org/wiki/SS-N-22

Looks like these missiles are at least 20 year old. How come there's no defence against them yet?

There are several already deployed defenses against these, and weapon systems similar to these. The basic chaff deployment systems, CIWS systems for close in defense, and several anit-missile missile systems can all be leveraged to provide defense against an incoming attack.

Additionally, the strongest carrier defense against weapons such as this is the size of the ocean, and their short(ish) range. You can't hit what you can't find, and/or what you can't get close too.

They're simply too fast and too agile, employing evasion techniques. The only defence is lasers and railguns - thus, we're getting lasers and railguns Real Soon Now™.

>inflicting a sharp defeat on the US in the pacific would crush public morale and push it into a negotiated peace

This would depend squarely on the context and circumstances. Historical precedents are of little to no relevance here.