The major airlines asking about your driving history, and asking you to disclose your infractions, is largely an integrity exercise more than it is a screen to keep people who make bad decisions while driving out.
Yes, a record with repeated speeding tickets can be viewed negatively, and larger infractions like Negligent or Reckless Driving are going to be noteworthy on an application. Certainly DUIs are going to be high hurdles, but not necessarily show-stoppers.
However...people make some elements of this into a substantially larger issue than it actually is. There is ample evidence of people with many traffic tickets (usually speeding) being hired at the major airlines, and not just in this latest hiring wave where there has certainly been a lowering of the bar across the board in terms of qualifications. Yes, it is one of many indicators of an applicant's personality and fitness for the job, but in and of itself a list of traffic tickets isn't going to derail an application. Obviously it is better to have fewer, and the further they are in the past the better.
What absolutely is a 100% killer is not being 100% forthright on your application about any tickets or infractions. Again, this is largely an integrity exercise, and there is also ample evidence here of applicants being offered CJOs at an interview, and then later having those CJOs rescinded because something popped up in the background check that wasn't disclosed on the application. The airlines absolutely, positively, will not hire someone they see as appearing to be anything less than 100% honest.
Whatever system the airlines use to check driving records appears to differ from the typical National Driver Register check that we can perform on our own records, and even the individual state checks that we can do to our own selves. Thus, "I got a ticket a long time ago, but it isn't on my record, so I'm not going to list it on my app" is a very poor path to take -- one which has bitten people in the butt before.
So, don't fret about a couple of tickets. Yes, two tickets may be a pattern, but I guarantee you nobody is not getting called for an airline interview over two traffic tickets on their application.