What unemployment is going to be interested in was the exact reason for the very last tardy. The one where your alarm clock didn't go off, and you were fired that day for that particular tardy. It was the proverbial last straw, right? So that's the one they will want to talk about. Did you know, had you been given enough warnings that you knew that if you were tardy again (for an AVOIDABLE REASON) you were in grave danger of being fired?
The thing is, in an at will state, the company does not have to follow its own handbook or rules, does not have to count this as one point and that as a half point and not count this at all as an approved absence. They can fire you for three or two or one medical reasons with a doctor's note for each and every one of them. But then unemployment is your recourse, IF you can get approved for unemployment.
If on your last day, you had started off to work timely and on the way there, your car was struck by a bus and you were rushed to the hospital in critical conditions (and I've seen this actually happen) and they fired you for being tardy anyhow (and I've seen THIS actually happen) there would still have been no labor laws broken. But you would, when you were finally able and available to work again, be able and very likely to be approved for unemployment benefits. Your tardiness was NOT something you could reasonably have avoided. Last tardy+medical excuse is usually considered unavoidable, and while your firing was legal, it will probably be possible for you to draw benefits.
But when you are hovering on around the edge of a termination, getting warnings and disapproval feedback of various kinds from your employer, and you don't take the reasonable actions to avoid having another tardiness incident, you know, the action of buying a new clock, having friends call to check on you, setting a cell phone alarm, etc. and you get another tardy because your alarm clock didn't go off AGAIN, then you are fired, with nothing illegal in your firing, and not a very good chance at all of being approved for unemployment benefits either.
You can get into the unemployment insurance case and argue and tell the adjudicator that you were being treated unfairly, that the company didn't figure your "points" for termination correctly because one of them was for a medical situation covered by FMLA, so they really shouldn't have fired you for the occasion where they did fire you, you should've gotten at least one more warning and shot at it. And you may try to show that you weren't being treated fairly because the company did not use the same standard to figure points and terminate with another employee that you knew of. So you didn't think you'd be fired because they didn't fire him. Or because of what they told you in your warning meeting. What they're going to ask you, over and over, is "Did you know that being tardy again was likely to lead to your termination?" And you're going to have to come up with some way to make "my alarm clock didn't go off" sound like a convincing reason you didn't make it, even knowing you were about to get fired. And all that just to get approved for unemployment. There's NOTHING here that is going to mean any way to ask for and demand your job back.