Before elaborating on why I say yes, let's look at the other side of the coin, because there's quite a bit that can be learned from that first. Here is why sending that E-Mail might not make sense:
You blew it. You suspected that. I'm going to just outright confirm that.
You could say that your ride blew it by being late, but you blew it by counting on that ride. Maybe your cousin is typically responsible and relying on your cousin was a very sensible approach, but that doesn't change the fact that this job opportunity appears to have been blown. That needs to just be accepted, simply on the grounds that it is reality.
Since you're asking "should I not waste my time", it sounds like you're not interested in spending time on something that is quite unlikely. And I am saying, this seems quite unlikely. That is the key basis for saying "no" for the question that was actually asked (even though, as my last paragraph a notes, I would actually personally recommend "yes").
Here are some of the reasons I see the chances as being so unlikely:
Being late is bad
No surprise here... We all already knew that.
You did get unlucky. Sometimes you can get lucky. Maybe you show up, and the other person is also late. Or maybe the interviewer doesn't ever learn that you are late for some other reason, like you show up and speak to a receptionist who doesn't report your lateness. Unfortunately for you, you did not get such luck this time.
A made, and was made right away. So we know this was noticed, and it did seem to matter.
Your response was bad; arguably terrible.
You said, “I mentioned to him my perfect attendance at my previous job last year, never missing a day, never calling out/sick and how I was almost always scheduled for opening shifts at my current and previous jobs.”
You claim that you are typically reliable enough to show up to work, sometime (on the right day). You don't completely miss any days. But you didn't say that you actually show up on time. You didn't say that lateness is out of character for you. Do you know what you did? Just based on the text that you provided, it looks like you actually managed to sidestep the issue of punctuality, completely failing to directly address that issue that you were asked about. So not only were you late, but you were evasive, and didn't fully demonstrate responsibility for an important issue.
A better approach (in my opinion) would have been something like “Something bad happened. This lateness is out of my character. I failed to be able to sufficiently compensate for the situation.” At least that would have shown you : recognize the screw-up, are willing to acknowledge it, claim this is unusual, and will take the responsible approach of at least addressing problems as best as possible. Unfortunately, that didn't happen, and there are situations where people who lack time-travel abilities just don't get second chances. It does look like this is one of those situations.
Your comment (shown below the question) shows your fate is sealed.
“halfway through the interview the district manager was mostly on his laptop and didn't seem to pay much particular attention to what I was saying. Looking at the computer more so than me” -- Ooo-ouch!
“In short, not pleasant.” -- Ooo-ouch!
“But the assistant manager however was actually giving me body language feedback and was actively listening to what I was saying.” Don't get much hope in this. The Assistant Manager likely reports to the Manager (who may have been unavailable for this interview) who likely reports to the District Manager. If the hiring decision is made by the District If Manager, it sounds like you're doomed.
If the hiring decision is actually typically made by the Assistant Manager, you're likely still doomed, because the assistant manager is unlikely to want to rebel against the district manager. If the district manager truly didn't like you, he may have allowed the interview to continue (in part just to avoid unnecessary conflict, and perhaps in part so the district manager could continue to further gauge the assistant manager), but chances are rather high that the district manager may have let the assistant manager know that the just-interviewed candidate was not a good match. (Furthermore, the district manager may have told the assistant manager why.)
So, are you doomed? Well, your chances of getting this job seem very, very slim, but there's good cause for hope.
It's time for me to explain why, despite all of the less cheery outlook I've provided so far, sending that follow up E-Mail is a good idea.
I've royally botched some interviews that I really needed in life. I'm old enough that I didn't have this website available during that time of my life. My life did have some challenges as a direct result of not handling interviews very well. But eventually I learned some things and my skills increased and now there are plenty of people who would like to be in the shows that I am currently in. So, life can get better. (Not always quick nor easy, but it can happen.) In my case, that required me learning, working, and continuing to try for positive things. Took a while, but it did happen.
You, on the other hand, are using this website, which can be a great resource. So you are doing a good, right thing. Keep that up, by continuing to try to do the right and positive things (like spending the effort writing a brief E-Mail).
Typically the best answer is "Yes", sending that E-Mail is something worth doing, because the theory is that if there is still a 6% chance of success, that 6% is more than 2%, and sometimes things turn out unexpectedly well. Since the cost to follow up is so inconsequential, then, yes, follow up. (In other words, overall I agree with the first paragraph Kat's answer on why it is worthwhile. I also agree with Paparazzi's highly-rated comment, because you basically want things to be as positive as possible, and it may be that your only conceivable chance involves further focus on positive things. The third bullet point of Makoto's answer, about a strength, provides you with the most promising strategy.)
Even if this apparent potential job opportunity was blown, just follow Kat's advice and try to forget about what would have been nice; instead focus on what is likely to be nice as you find and pursue another opportunity.