That's why I'm positive that Beverly was lying and she was no longer attracted to Picard and was creeped out by him and doesn't want to be in a relationship with him, while telling him about Jack will guarantee she's tied to Picard more or less for life. So she purposely takes an isolated series of events out of context (note that every single incident she mentions is Romulan related, taking advantage of the chaos in Picard's life after he killed Praetor Shinzon, than anything actually representative of Picard's usual Starfleet life, much less the 15 years he spent quietly in France) to demonize Picard rather than just straight up admit she doesn't want to live with him or be in a relationship with him. People in real life do this all the time, demonizing someone to get out of having anything to do with them while making themselves look heroic for dumping/rejecting someone.Per Beverly’s line of thinking, Jack was only going to be in danger if someone found out he was JLP’s son. Since only he and his mom knew, I would think he could show his face anywhere because no one knows who he is and it’s been so long with zero contact, no one would suspect such a thing.
Unless it’s acceptable to go around using your medical tricorder to scan strangers’ DNA in the 24th century, then he’s got a problem.
The smoking gun so to speak is Beverly ended up putting her son in danger on the most lawless areas of space for her self-important frontier medicine gimmick, to the point Jack needed to turn to crime to survive (and per the math he just turned 21). She can't have been that concerned about his safety, she doesn't even seem to care about his seeming lack of post-high school education.
For what it's worth, real life Chinese director Zhang Yimou hid at least 2 children for some years to hide the fact he'd violated China's one child policy not that long ago. So Beverly totally could've gotten together with Jean-Luc and hidden their son's heritage if she wanted to.
The bottom line is, she didn't want to, because she doesn't want anything to do with Jean-Luc for reasons beyond her exaggerated claims of the danger around Picard (which she purposely exaggerated to make herself look heroic). The real reason is all but stated between the lines when Picard mentioned she refused to communicate with him further, the implication being if he persisted she'd be filing restraining orders, etc. In real life, that type of action is reserved for stalkers who won't accept no means no, and for Crusher to put Jean-Luc in that position where she could realistically accuse him of that speaks volumes about what she really thinks about him. She likely grit her teeth and put on a show to even work with him this past episode on the Titan only because lives were at stake.