Picard's presence on the covert mission is explained more than any of them. It's said he's the only active Starfleet officer experienced with the theta-band carrier wave nonsense, they'd been detecting on Celtris III, that'll be used to deliver the metagenic weapon. There were like 2 others, both out of service.A thought on Picard's involvement, going with the idea that Starfleet doesn't actually HAVE a group to handle this... he was put on the mission just because he was present and he needed to be taken off the ship. Starfleet wanted Jellico to handle the situation. Having Picard remain on the Enterprise would have certainly caused even more issues, so... they needed a body for the ground mission and Picard needed to go somewhere.
It's why the Cardies used it as a lure. It's not weak writing imho, if they're intentionally targeting him from intel, with something they figured out only he knew about, & designed a trap for him personally, so Starfleet would have no one else to call up for it, the goal being to grill him over the defense plans for Minos Korva, (where they were planning their incursion)
Him being there was because he's suspiciously the only expert, Crusher because of her medical expertise, which would be handy if bioweapons are in play & Worf for just being good tactically, & that they both are already on hand, under Picard... but why Picard? because he was the target. It was always going to be Picard, whatever the ploy.
Now, why Nechayez couldn't see that it was an obvious trap, that suspiciously targeted one guy in the whole damn fleet, is a good question. I envision two explanations. #1, she's an idiot, who's bloody dumber than Kennelly when it comes to Cardassians... OR the more likely #2, she knew it would be a trap, & sent Picard anyway, knowing he was never coming back, but considered it a necessary evil, on the off chance there WAS a bioweapon in play. Either way, both Picard & Jellico are proven better strategists than her any day of the week.