The criteria for statement C (which OpenAI targeted) only require a C-infinity smooth force, whereas the paper considers constructions of the same kind as the OpenAI one but with a real analytic force. All real analytic functions are C-infinity smooth, but not vice versa [2]. Wikipedia says the Navier-Stokes problem for real analytic forces is still unsolved, so the question of whether a similar approach can be used there is presumably of some interest, but it doesn't appear to affect whether the Prize criteria have been satisfied and the paper doesn't explicitly make any claim that it does.
Re the problem still showing as "active": The Clay Institute don't consider problems solved until the solution has been fully digested by the mathematical community and is well established. In the case of the Poincare Conjecture that wasn't until 2010, 7 years after Perelman put his papers on the arXiv. Nobody is expecting them to mark Navier-Stokes solved any time soon.
Your comment was the first I've heard of that paper, BTW, so thanks for that. However, that's another indication that this probably doesn't bear on the Millennium Prize criteria. The lead author, Peter Constantin, is a leading expert on the problem and the Clay Institute's page on it has a video of a lecture by him. If he had come out and said OpenAI hadn't solved the problem as stated I'm sure there would have been a lot of noise about it. The paper is dated 17th of September so presumably there has been plenty of time.
[1] https://www.claymath.org/wp-content/uploads/2022/06/navierst...
[2] https://en.wikipedia.org/wiki/Non-analytic_smooth_function