This is a bit nuanced and complicated. You're right in spirit of course.

Technically, those were the nuclear weapons of the Soviet Union. After it broke up, operational control of these weapons remained in Moscow as per https://nucleardiner.wordpress.com/2022/02/06/could-ukraine-have-retained-soviet-nuclear-weapons/

So Ukraine had physical possession, but they couldn't have turned them on from day 1 of independence. And if Ukraine had refused to return them, it seems it is an open question if they could have circumvented the security measures or not to gain control over them.

Ironically, my understanding from https://www.tandfonline.com/doi/full/10.1080/00396338.2015.1026091 is that part of the reason Ukraine agreed to give up those nukes was in return for having not only security assurances, but to have those assurances extended to Crimea. This can be viewed and exchanging the nukes for retaining Crimea.

Considering what we know now... that might not have been the best deal. This almost has me asking, why not both? (Both NATO membership and nukes)