How do I tell a professor who tells me he'll send in a recommendation but doesn't that I don't need his letter anymore?
If you don't feel angry at your professor, you could drop them a note once the firm deadline has passed, so that they avoid wasting time on writing a letter that's not needed anymore, or they don't feel bad about never having gotten around to doing it. I could imagine doing this with some colleagues whom I know are very busy.

In this case you could write something like "Dear professor, I'm writing back about the letter of recommendation that I originally requested from you on [date]. You had agreed to write this letter, but I have not received it. I just wanted to let you know that the deadline for submitting letters has passed, so it doesn't make sense for you to work on this anymore. Instead, I have asked [other prof] to write a letter explaining the situation."

If you will be working with this professor again and would like to set a boundary, you could add something like "It felt stressful for me to make this alternate arrangement without having a response from you about the original letter. In the future, it would be really helpful if you could respond more quickly to email about this kind of matters." Or something like this, although that's hard to word.

On the other hand, if you feel indignation at your professor, as seems to be the case, then I don't understand the purpose of sending a further email at all. What good would it achieve? If the letter of recommendation arrives at some later time, you can simply explain that the deadline has passed so you can't use it, but thanks nevertheless.