How exactly was Kathy Morningside, and to a lesser extent her son Frank, planning to get away with bombing the Miss United States pageant, when they did such a lousy job trying to cover up their involvement and motive? Kathy carelessly licked the envelope before sending the fraudulent "Citizen" letter threatening the pageant, she spent hours writing other hoax letters right in her office with the door unlocked, she lied to the FBI about being fired, she helped Hank switch the winner's crown for the rigged one in plain sight sight with numerous witnesses all around, and Frank chose the worst possible place to press the remote button that triggers the explosive mechanism in the crown if he didn't want to get spotted. Had they succeeded with their plot to blow up the next Miss United States winner, they would've been caught within minutes.
Not exactly; nobody would have been looking for another suspect if the true Citizen hadn't been caught, Frank's parentage can't be that well-known if Vick had to alert the FBI to it, she could legitimately claim that she hadn't thought her being fired was relevant to the case, and she could have argued that the crown had been rigged before she switched it.
Also? She's kind of delusional. People who jump from, "I'm being fired," to "Therefore I have to murder an innocent girl on national TV," are not known for their ability to rationally and correctly plot out the consequences of their actions.