What would be the best way to convince a professor to waive late submission penalties?
Professors will say no by default to all such requests, but a lot of the time they secretly want to say yes, so you need to help them help you by providing all the relevant context, being logical and reasonable, and generally convince them that you are worthy of their attention and understanding.
Here is my attempt at a draft email.

Dear Professor xyz,
my name is ZYX, I’m a student in your Programming for Programmers class. I submitted the first class project one day late on February 17 and as a result you imposed a 20% late submission penalty, which reduced my project grade from 18/20 to 14/20.
I know that this penalty is in the course policies and accept that there is a good reason for such penalties and that you have the right to decide such things. But I wonder if you considered that such a high penalty may be unreasonably harsh? I am an excellent student and have worked very hard to maintain an overall GPA of A+ in the program. Because of the penalty it seems almost certain that my final grade for the class will end up lower than what my performance would merit without the penalty, which in turn would lower my overall GPA and future academic and career chances. The consequences seem disproportionate relative to the one time mistake of submitting an assignment late by just one day.
I therefore want to ask respectfully if you might consider waiving or at least reducing the penalty for this late submission.
I do sincerely apologize for the late submission, and for not contacting you in advance of the deadline to request an extension (something that in hindsight I obviously should have done when I realized I was going to be late). Normally I am very good at keeping up with my workload, but back in February, because of a research internship I was doing at [name of company], which required me to commute back and forth between [name of city 1] and [name of city 2] and took up a lot of my time and energy, my commitments ended up piling up in an unexpected way, which caused me to fall a little behind and miss the deadline. I nonetheless worked diligently to catch up and ended up managing to submit what I think you’ll agree is a high quality project the next day.
Thanks for your consideration. I hope you understand my anxiety and do not think less of me for making this request. In this difficult period when students like me are facing an uncertain job market and other unusual difficulties related to the COVID-19 crisis, we could use a bit of leniency.
Regards,
ZYX