In the companies I've worked for, employees have always been given the choice of how to communicate to others that they were leaving. In the majority of cases, this meant the employee writing their own goodbye message on either email or the company intranet, some people would just announce it during a meetup with their own team and occasionally someone would ask HR to make the announcement instead.
In all cases, with people saying their own goodbye, they would include a few kind words and usually something about where they plan to go from there. This made them sound less like "someone just got axed" and more "someone has different future plans", even in the cases where the company let them go.
It also made any follow-up messages from HR about practical concerns (like who takes over which part of their job for now) much less surprising and everyone already felt "in the know", even though we don't actually have any idea why people left.
This personal touch makes quitting/getting fired feel much less like "the dice decided it's your time to go" and much more "there was a good reason for this and the person will be okay". (Even if it's not, because again, we don't really know).
Of course this only works for cases where people aren't being removed from the premises immediately. I've only experienced one case where someone was terminated for cause, and for that one the owner of the company called an all-hands immediately after (fairly small company) to inform the others that so-and-so was removed from the company effective immediately. In that case, it is a shock, but it is also very clear to everyone that this is A) an unusual occurence and B) not a random decision
So you still don't feel like you might be let go at any time.
Of course, these behaviors work best in a company where people aren't let go at any time and then immediately removed the premises. If that's how your company operates, I don't think any kind of sugarcoating is going to really remove the feeling you might be let go at any time, because that is the truth of the situation. My only advice at that point would be not to work for a company like that, the best way to not be reminded about the situation you're in, is not being in that situation in the first place.