And Bill Bradley was not the only politician to confuse Gorbachev, the leader of the Soviet Union, with Yeltsin, the leader of the Russian federation. Gorbachev was not talking about Russia per se, but the Soviet Union. That government collapsed and disappeared. There was no détente between the USSR and the US anymore, it was between a lot of sovereign governments, one of which included Russia, and the US and all those European governments. So the situation changed quite drastically. The Russian government was a completely different political entity that simply inherited many Soviet roles internationally, because it had dominated all of those now independent nations that emerged from the USSR. Russia inherited (albeit not actually by informal agreements) the Soviet seat on the Security Council. It also inherited debt and assets, but not all of those nuclear weapons in Ukraine. So a deal was struck WITH RUSSIA to secure those weapons, turning many of them over to the safekeeping of Russia, in return for a commitment by the Russian government to honor the sovereignty of the Ukrainian nations--the Budapest memo in 1994. Putin reneged on that agreement exactly 20 years later, but the US, UK, and Ukraine still considered it in effect.
Note that the Republicans, which included Bill Bradley, gave nonbinding verbal assurances to the Soviet Union, not Russia, that NATO would not expand. Those verbal assurances were never formalized and had never bound the actions of subsequent US administrations. After the dissolution of the Soviet Union, the situation changed drastically. All of those former satellite nations in the Warsaw Pact and former captive "republics" within the Soviet Union wanted NATO membership. Why? If you need someone to tell you why, you are hopeless. But I don't think you need to be told.
NATO didn't move eastward. The Warsaw Pact and Baltic republics moved westward, as quickly as they could. They wanted NATO for military defense, and Europe for economic development. Ukraine wanted that for the same reasons, but that never happened and wasn't about to happen when Russia invaded Ukraine in 2014. Bradley died in 1997--before Putin came to power. He never had a chance to even address the situation in 2014, let alone 2022. Russia did not invade because of NATO. It invaded, because Putin feared seeing Ukraine slide closer and closer west, just as the former Warsaw Pact nations had. Putin is a Russian ultranationalist who believes that Ukraine belongs inside of Russia. The US, western Europe, and NATO did not entice eastern European countries to join NATO. Those countries enticed NATO and Europe to let them join. Had they not been allowed to join, Russia would not just be threatening Ukraine today. It would also be threatening to annex the Baltic republics and install puppet regimes in former Warsaw Pact nations. Russia has validated their reasons for wanting to join NATO in the first place. Instead, a Russian dictator saw a Russian-speaking nation on its border suddenly developing a healthy western-style democracy, which threatened his autocratic grip on power.