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.
There were actually aircraft designed by the US to match the Tu-160
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.
The Valkyrie was wound down because missile technology made it obsolete before it was even done. The USA instead focused on the supersonic B1. And even that project was almost cancelled as obsolete if Reagan hadn’t had intervened, and gone for a low flying B1B instead that had much less chance of being shot out of the sky before dropping its bombs. To be honest, even the B1B isn’t that useful, ballistic and cruise missiles do the job much better. Even Russia doesn’t fly its supersonic bomber very often, flying the less expensive to operate TU-95 a lot more.
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.
The main reason the XB-70 and B-1A were shutdown was yes missiles advancing, but mostly price.
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.
Valkyrie was a different class of bombers, one that never came to be. Rather than the Blackjack, the Russian answer to Valkyrie was actually the T-4. The SR-71 was a spyplane, but there was also to be an SR-71-class interceptor meant to shoot down incoming Valkyrie/T4-class bombers. That entire field of mach-3 bombers and fighters was stopped by advanced in missile tech.
