As to what the FO button does, it cuts off whatever call the number you were dialing is on and connects your call instead. So you had better be sure your call is is important enough to potentially cut off a General Officer.
At least that's how it was explained to me back in 2003. Never had occasion to use the priority modes.
It's more that on systems with a circuit switched network there are typically fewer outgoing/incoming lines at a location then there were telephones, so you can get in a situation where a choke point in circuits can mean that you can't make a call because they're all busy. FO will force hang up other calls to make sure yours can go through. That's very likely to hang up other, unrelated calls in order to get through, rather than the person you're calling (it will hang up their call too if they're currently on a call though).
In the late 90s I was with a group of people and we were trying to make a call. The line was busy, had been for an hour. We assumed it was the teenage daughter on the phone. One in the group took the phone, dialed a lot of numbers, then asked for the number we were trying to call. He punched it in and handed the phone back. The call went through and the confused teenage daughter answered - confessed because her call was interrupted. Everyone in the room was amazed and he went back to reading as if nothing happened.
I don't know what went on, but I can only imagine it was related to FO. My thought was he punched in to somewhere that would let him use a non normal button by way of numeric entry and connected the call - but I don't have a clue.
yes, it sets your priority for throwing others off of interswitch trunks and keeps others from throwing you off, as i understand.
traditional phone network would give a fast busy or an "all circuits are busy" message if all lines between two switches are busy. the special buttons let you assign priority for your use of those lines. F0 is just max priority.
...and now imagine how likely "all circuits are busy" would be in a network were half of the infrastructure was freshly nuked. A dedicated button for "give me a line, I'm the president!" doesn't seem that far fetched anymore. Who knows, if they spend enough time pondering communications availability in such dire circumstances someone might invent an internet!
Modern phones could do with an "FO" button for an entirely different purpose. Don't like the current call... hit the FO button... let them know. In particular a disturbing practice I noticed in Australia. The bank calls, I pick up and a recording: "this is XYZ bank, please hold an operator will be with you shortly". A golden opportunity for an FO button. It could work like an email spam button Enough FO's and the number gets auto blocked.
It already exists on many telephone service providers in some countries. It doesn't have a specific button, and it isn't standardized across providers. But on one provider in the United Kingdom, for example, it is 14258 then star twice to blacklist the number that last called. (On another provider, just to show the variation, it is 1572 then 1.)
The downside of a single button would be that even with the existing systems it is all too easy to accidentally blacklist the wrong people. I know someone who was accidentally blacklisted by one of xyr elderly friends. It caused all sorts of interesting fallout until the two of them had the opportunity to speak in person.
One can imagine what a single button could do in the hands of small children. And, indeed, we have known for a couple of decades what anti-spam buttons can do in the hands of people who (say) decide to "junk" every mail message after reading it.
For Australia, see rather the Do Not Call Register, which is the better option as these blacklisting systems are usually restricted to a couple of handfuls of numbers.
https://upload.wikimedia.org/wikipedia/commons/b/b2/AUTOVON_...
Now we have the answer to one of lifes burning questions, what does the FO button do?