First, according to Gorbachev, the topic of not allowing former soviet states was never discussed in 1990, so there was no oral assurance given on that subject. Gorbachev in a 2014 interview:
The topic of ‘NATO expansion’ was not discussed at all, and it wasn’t brought up in those years. … Another issue we brought up was discussed: making sure that NATO’s military structures would not advance and that additional armed forces would not be deployed on the territory of the then-GDR after German reunification. Baker’s statement was made in that context… Everything that could have been and needed to be done to solidify that political obligation was done. And fulfilled.”
Gorbachev continued that “The agreement on a final settlement with Germany said that no new military structures would be created in the eastern part of the country; no additional troops would be deployed; no weapons of mass destruction would be placed there. It has been obeyed all these years.” To be sure, the former Soviet president criticized NATO enlargement and called it a violation of the spirit of the assurances given Moscow in 1990, but he made clear there was no promise regarding broader enlargement.
Russian President Vladimir Putin has made it well known his antipathy towards NATO, claiming the Alliance took advantage of Russian weakness after the collapse of the Soviet Union in violation of promises allegedly made to Moscow by Western leaders. Steven Pifer argues that no such promises were...
www.brookings.edu
Common sense should tell you that since the NATO Charter states every European country is eligible to join NATO if it applies and meets certain criteria, so it would have been impossible for NATO to have made such assurances, and the fact that there is no contemporaneous record of such discussions would tell an honest person that Putin lied.
Before Russia's 2008 invasion of Georgia, there was little interest in Ukraine to join NATO and little interest in the US to see Ukraine in NATO, but after Russia's invasion of Georgia, which Putin tried to justify with the same lies you has used to try to justify his two invasions of Ukraine, there was a sudden increase among Ukrainians in joining NATO, and Ukraine applied and Bush supported that application, but sadly, too many Europeans were still deluded into thinking peace with Russia was possible. Had Ukraine been admitted in 2008, there would have been no 22014 or 2022 Russian invasion of Ukraine.
NATO was not out recruiting nations to join; it was responding to applications from former soviet states that didn't trust Russia according to its Charter.
Since NATO has posed no threat to Russia's security since 1991, the only reason Russia has to object to former soviet states joining NATO is that it prevents Russia from attacking them, and that is the reason why it is important for them to join NATO.