My professor agreed to give me a LOR. However, despite many reminders, he has yet to send it in. I called him up a week before the deadline and he was annoyed at this. I then considered asking someone else but he emailed me asking me for supplementary documents. Again there has been radio silence from his side after that. Now, the deadline has passed. I think professors can still send in LORs and I’ve asked another professor to write a LOR for me after explaining my situation. My question is do I email my earlier reference to tell him I don't need his LOR anymore? How do I word this email? Should I show my indignation at being taken for a ride?
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.
I'd suggest avoiding anything that could cause conflict, and anything that feels like you are wanting to vent. It will have been extremely frustrating for you, but telling the professor that will do no real good and risks creating enemies where they are not needed.
If you feel you wish to contact them, it might be worth sending a quick email to thank them for helping you with this, and apologising that it is no longer needed (you do not need to say why, or that the deadline has passed - you are just avoiding them wasting their time writing the letter now). It may seem odd to apologise when you have done nothing wrong, but from their perspective - you have just badgered them repeatedly/added stress to their life and are now saying it was for nothing. It is not saying you were in the wrong, it is just maintaining a relationship with them.
If you don't care for them at all and don't see a bad relationship with them causing you any problems in the future (note: I've found I'm still in touch with most of my old professors, or people who know them/worked with them, in some form or another) - just don't say anything.
It might feel like they owe you an apology, or that they should recognise how they stressed you out - but that's really just for personal feel-good. It doesn't help your professional career at all to chase that.