There were actually aircraft designed by the US to match the Tu-160. See the XB-70 Valkyrie, which was a complete failure that never gave suite to anything. So yes, the US never matched the Tu-160, despite trying.
At the risk of “arguing with someone on the internet”, I think you may have your chronology backwards, since it seems the Tu-160 was designed to match the US aircraft you mention.
The Valkyrie was a late 1950’s to early 1960’s design with physical prototypes existing and flying by 1964.
The B-1A design began in 1965, with the first flight in 1974. This would have been the closest western analog to the Tu-160, since the B-1B derivative had a different focus - low-altitude terrain following - since it was believed that the high-altitude/high speed designs like the B-1A were vulnerable to newer Soviet missile designs. Stealth was also becoming increasingly important to US command due to the same concerns about missiles.
The design competition that begat the Tu-160 began in 1972, with a first flight in 1981. I’m not saying it’s not a great plane - by all accounts it is very capable - but US designs were not done in response to it.
War isn’t a pissing contest to see who can have the fastest X, it is rather one where you find a bunch of most effective tools for the job.
Low flying approaches and cruise missiles were strongly reconsidered at the the because the MiG-31 had look-down shoot-down capability. We don't know if it would have been a better or worse option, we're dealing in counterfactuals.
Russia has few uses for the Tu-160 anymore. It's main use was to penetrate hostile airspace of near peer opponents. Since the fall of the USSR yes there is not much more use for that.