But, that's all a bit heavy for what this is trying to accomplish, which is certainly an understandable goal.
However the proper way to express anger in our enlightened business environment is passively and non specifically. "I'm unsure that this project is accomplishing it's goals." or "the team is having trouble with the vendor you chose" and then you schedule a 1:1 meeting to talk about it over a warm beverage where you adhere to a 10:1 complement to criticism ratio.
Both of those statements are completely reasonable. Don't conflate directness with anger, berating, personal attacks, etc. It's entirely possible to be direct without attacking people.
This sounds like you're saying anger is never okay. Is that true?
Because I can think of many instances when there's something more important than "not being angry", and anger may be necessary to achieve that goal.
I also don't follow your prescription for roundabout criticism. I've worked successfully with many teams where we were much more direct, and we were very pleased with our outcomes, and fellow team members.
Allegedly, the problem was that I hadn't apologized that they were having a bad experience with a service that we offer.
Yesterday I was sent one from a partner saying that some files we were to transfer to their FTP server (yeah, I know it's 2017) were missing. So I logged into the server manually and checked -- the files were there, created at 5 AM, just like they are everyday when the cron job that generates them was run. So I started my response by saying, "I just checked the server manually and the files were there. Why do you consider them missing?"
But then I started to think...would this reply come off as if I were questioning their intelligence? They must believe that the files actually aren't there, or else they wouldn't have sent me an e-mail asking. The same process has been running everyday since February of last year, however, and presumably they had never failed to find the files before yesterday.
I just sent the e-mail anyway, but it is interesting that I've been conditioned to question myself over a very innocuous reply to make sure my tone was appropriate, as to not risk offending the other party.
For example, there was a multiple day e-mail back-and-forth debating XML namespaces for SOAP requests on another related project. My code generated XML with different namespace labels than those in their examples, so they told me it was "incorrect". I had to send them the specific section of the SOAP spec to illustrate that the labels can be anything as long as they are set to the appropriate URI -- they're labels. They didn't believe me, thought that the differing namespace labels were what were causing the requests to be rejected on their end...until a few days later when it turned out that it was a completely unrelated bug in their system.
That's a good habit to get into, particularly given how many of our usual non-verbal cues don't come across in email.
I often think to myself "I'm about to send a mail to X hundred/thousand people; it's worth my taking a couple extra moments or even minutes considering exactly how it might come across".
The problem with anger over text, especially email, is that the angry message sits there and stews and stays angry at the recipient for hours or days. And they have lots of time to stew themselves and consider their response and go over lots of counter-anger thoughts in their own mind, before sending something back.
In-person anger can come and go quickly; it gets resolved; it doesn't continue for hours and hours and wear people down. Especially the way men interact (in some subcultures), anger is just a way to communicate a sense of urgency or import, and can quickly reverse into relief and camaraderie once it's acknowledged. (Of course this isn't universal, but it is a style that can work depending on context and culture, and if it's not overused).