I don't think it would be weird if you present your thoughts in an earnest and sincere way. Setting up an all managers/supervisor email could be an easy solution? You seem well intentioned, so that's a good start.

For me, a button on a website or app would be ideal. It avoids the difficulty of trying to find the right words use.
A simpler option could be a pre-written email or text message template (if email or text was possible).

Everyone is different, and talking to them is usually the best way to fund out their needs. I think offering your suggestions is a good start, but don't try to pressure them into picking one. Maybe they already know what they want, or maybe they don't know but could be inspired by a similar solution.

Approaching the problem mechanically is perfect, and saying 'well if X doesn't work what about Y or Z' is the right approach. If you lead with that he may have some refinements or suggestions. When he hears that you're open to this, being helpful, I'm sure he'll have ideas as well and you can figure something out.

Depending on what he's experiencing, he may be able do a one word answer, and then he can give you a word that he can rehearse that's code for 'Yo I'm calling out for today, but can't actually get those words out'. If he's able to give a yes / no, then he also could call you with silence, but then you see that he's the caller and say "Hey [name] I don't hear anything, are you in a shutdown and trying to call off?" and he can respond Yes (listening comprehension is often less affected). There also may be tech solutions, like Google Assistant making calls for people, that might be able to be pre programmed to make the call for him.

Nice that you're putting effort in, that alone is a huge help. I'm no expert in autism or even general cognitive disabilities, but your suggestions are pretty good, and not weird at all.

If he has a smartphone, there should be accessibility settings that let him use text to speech during a phone call, and I know on iPhone you can save phrases to reuse later. If you decide to try this method I would have someone help ensure it’s set up properly and he knows how to use it; which would also be true for someone who had a throat surgery or something, that shit is buried in menus if you don’t know what to look for or how to use it. And that’s definitely not something you want to try to figure out in the middle of a shutdown.