Rather then carriers, the Soviets had these big, nuclear powered missile cruisers. I guess the idea was to fire supersonic, surface skimming missiles at enemy surface vessels.
The idea is interesting, and it sounds like a means to challenge a carrier group without a carrier themselves. But being a surface vessel, they must have been vulnerable to aircraft. How would it have played out?
Cat & mouse. As you suspect, they could not reasonably hope to handle a carrier group on equal footing, but there were few carrier groups and relatively many missile ships, so the hope was that one could manage to get into range and inflict damage by exploiting the fact that even a carrier has a hard time scouring all of the large, empty sea around it. They also weren’t alone - they were part of a multi-layer defense that included submarines and heavy missile bombers, the latter being where the real strength was. The combined effect provided for a reasonable level of threat to enemy carrier groups that might try to move within operations range of mainland territory.
It was a decent response to the hopeless situation that was the total naval dominance of the USA. Offensive capability against that overwhelming strength was essentially futile, but with the assistance of less-mobile assets, a credible defense was possible.
They were pretty vulnerable to air attack, and most navies (USN included) have phased them out, in favor of smaller, more capable ships (destroyers and frigates).
Face it, for the cost of one cruiser, you can have 5-6 destroyers-more firepower, and more dispersed/less risk of loss.
Phased out? The US Navy currently has 22 cruisers in active service.
And the current USN “destroyers” aren’t much smaller than the Soviet cruisers. The Arleigh Burke class destroyers are 155 meters long and displace 8-10k tons. The Kirov battlecruiser is significantly bigger, at 255 meters and 28k tons, but the Slava heavy cruisers were more comparable at 186 meters and 11k-12k tons.