Kingsley's patronus was not the only one that spoke.
In Chapter Seven of Deathly Hallows Mr. Weasley's patronus spoke:
They all saw it at the same time: a streak of light that came flying across the yard and onto the table, where it resolved itself into a bright silver weasel, which stood on its hind legs and spoke with Mr. Weasley's voice.
"Minister of Magic coming with me."
In Chapter Nine Mr. Weasely's patronus spoke again.
Then Hermione shrieked: Harry drew his wand again and spun around to see a silver Patronus soar through the drawing room window and land upon the floor in front of them, where it solidified into the weasel that spoke with the voice of Ron’s father.
"Family safe, do not reply, we are being watched."
Earlier in Chapter Nine Hermione states that she thinks she can do the talking patronus thing:
"Can you do that talking Patronus thing, then?" asked Ron.
"I’ve been practicing and I think so," said Hermione.
There are also other instances where a patronus is used to send a message, but we only see it being sent but not delivered. For example, in Chapter Eight of Half-Bloood Prince Tonks sends such a message to Hagrid (which got intercepted by Snape):
As Harry swung the cloak back over himself, she waved her wand; an immense silvery four-legged creature erupted from it and streaked off into the darkness.
"Was that a Patronus?" asked Harry, who had seen Dumbledore send messages like this.
"Yes, I'm sending word to the castle that I've got you or they'll worry. Come on, we'd better not dawdle."
Similarly in Chapter Thirty of Deathly Hallows McGonagall uses this method to send a message to the other heads of house:
She marched toward the door, and as she did so she raised her wand. From the tip burst three silver cats with spectacle markings around their eyes. the Patronuses ran sleekly ahead, filling the spiral staircase with silvery light, as Professor McGonagall, Harry, and Luna hurried back down.
This is presumably what Dumbledore was using as well in Chapter Twenty-Eight of Goblet of Fire, though it is not referred to as a patronus there:
He raised his wand into the air and pointed it in the direction of Hagrid's cabin. Harry saw something silvery dart out of it and streak away through the trees like a ghostly bird.
Since we don't see the delivery of these messages, we don't see the patronuses speaking, but that is presumably how the messages were delivered.
As for why this method wasn't used in various instances, let us examine each of your examples in turn:
Harry could have warned his past self in the Prisoner of Azkaban.
In Prisoner of Azkaban Harry was a novice at magic in general, and at the Patronus Charm in particular. He had only just learned how to do it at all; sending talking messages with it is much more advanced magic which Harry wouldn't have known at the time, and he probably didn't even know of the existence of such magic.
Snape could have used his doe patronus to speak directly to Harry in the Forest of Dean. Snape's patronus could have told Harry about the sword of Griffindor. He could have told Harry how he was acting under Dumbledore's orders. He could have saved Harry many troubles.
As described in the above quotes, the patronus speaks in the voice of the caster. If Snape would have used this method of sending a message, Harry would have recognized his voice. At that point in the story Snape was still maintaining his cover and it was not yet the time to tell Harry the truth.
Sirius could have used a patronus to speak to Harry several times.
While Sirius might have technically been able to use this method of communication, he would not have had much opportunity to use it with Harry. For the first two years or so after he escaped Azkaban, we don't ever find that he had a wand so he would have presumably been unable to cast the spell. In Order of the Phoenix when he might have had a wand, it would have been too risky to use this method. Anyone around Harry at the time would be able to see the patronus and hear Sirius's voice. This would likely have alerted people to the fact that Harry was in contact with Sirius, something which Umbridge was already trying to prove throughout the year.
Members of Dumbledore's Army could have informed the aurors and the Order of the Phoenix about the battle in the Department of Mysteries.
Indeed, this is presumably what Dumbledore meant in his statement to Harry in Chapter Thirty Seven of Order of the Phoenix:
I should explain that members of the Order of
the Phoenix have more reliable methods of
communicating than the fire in Dolores Umbridge’s
However, this magic is probably not something that the students would have been able to do. As mentioned above, more than a year later Hermione wasn't even certain that she could do it. At the time, it was simply not a possible option for them.