Wikipedia notes both the Spartan and Sprint missiles as having nuclear warheads. That was reasonable-ish, since Wikipedia also notes them being cold war-era anti-ICBM weapons. Less bad to have your own interceptor nukes going off "near" your city than to have enemy nukes scoring direct hit on it.

In contrast, modern hypersonic weapons have plenty of use cases where they'd be fitted with conventional warheads, and used against targets like US Navy ships.

There is plenty that could go wrong if USN ships mounted nuclear interceptor missiles, ready to launch on a moment's notice...