Not happened to me yet, but it would be awful indeed!
!!! Critical Bug at the start of the game

I'm also really hoping to see this fixed quickly... a player flagging w/out moving is a HUGE handicap to the player opposite.
It's happened twice to me; both times I was in adjacent position and reaped the free +20 (after all, someone had to take it :-) ). One game was won easily, in the other game, the remaining 2 players (rightfully) coordinated to level the playing field a bit.

What to you suggest, BabYagun? The abort button was being abused egregiously. It was intended as a "this guy did not show up" button, but ended up being an "I don't like this matchup" button. Too many abandoned games. When we took it away, I expected to see a TON of complaints. But we've had 1/100th the complaints we had about abandon being abused. At least with no abandon button, the person who disconnects and flags is punished from a rating standpoint.

IMO, the biggest problem is the "freebie +20" king. In many cases, one of the other player's first move will have left them able to grab the king uncontested. Perhaps if a player flags without moving, their pieces just vanish from the board.
This still puts the player opposite at a positional disadvantage... perhaps a token +5 points to the player opposite as a measured counterweight to this disadvantage?

It is irritating when after start someone quits without a move. I agree with Kevin: the opposite player has unfair and large disadvantage but I think it can't be compensated for a few points.
I recommend automatic abort in this situation (so still no need to have Abort button).
While matching up there is a Cancel button to abandon the game without annoying others.
So the abandoners should be punished somehow. Eg. should count their abandoned games and allow the players when start their games to choice opponents with last-90-days-abandonments-% just like at daily chess there is the "Avoid Time-Outs" setting. Fair players would be best protected by this, I think.

@ignoble, You do not want to add the "Abort" button. And you like that "the person who disconnects and flags is punished from a rating standpoint". It is possible to satisfy both these conditions. You can decrease the rating of the disconnected player (by placing him/her to a virtual 4th place in this game) and then abort the game. Why not?
If it is not enough then add him/her to some "graylist" and do not allow to start a new game for 30 (10? 20?) minutes.
Another solution is what @kevinkirkpat just offered. "Remove the abandoned king and add +5 to the sandwiched player". But I do not like it. The sandwiched player is most likely to lose. But any compensation (+5 or +15) is questionable and also not fair. The fair solution is to abort the game.

More temperate solution (considering that the server can have a "disconnection bug" and not all the players have a decent connection): If a player disconnects 1 time in 8 (6? 12?) hours, just abort the game and graylist the player. If he does it for the 2nd time then decrease his/her rating and also graylist.

it happened to me once to have "three-players chess" but I think it's not so common and if it keeps being this rare, I think the actual solution is not an issue

Abusing abort was a real issue some time ago, as it is evident from topics like this and this. My understanding is that the developers changed the default to offer a quick relief to a bad situation, without elaborating the system any further. Admittedly the situation is much better now. Personally, if an early timeout ever happens to me, I would rather resign immediately than having to play a 3-player chess game in a 4-player chess board. I care more about having a good time than about losing some rating points. I'll surely regain my points in the very next game!

It happened to me 3 times during the last 10 games. At 2 of them were one after another.
I also spectated a game of one of the top players, I joined spectators in the middle of the game and noticed that one of the players had the gray pieces on their initial places. I asked in the chat about that and got the answer: "He did not start".
I think the developers have some statistics, or can gather it, if needed.

One option if someone doesn't move at the start of the game would be to automatically replace that player with someone else.

Another solution is what @kevinkirkpat just offered. "Remove the abandoned king and add +5 to the sandwiched player". But I do not like it. The sandwiched player is most likely to lose. But any compensation (+5 or +15) is questionable and also not fair. The fair solution is to abort the game.
Wouldn't that be the same as having an abort button?

@happyyellowhandball, "abort + punish" is not the same as "abort". (You just quoted a part of the text, this way the suggestion loses its context and a part of its meaning.)

Wouldn't that be the same as having an abort button?
It wouldn't. If there is an abort button, anyone could use it and for whatever reason. If there is automatic abort, it is triggered by only when someone quits without a move.
I think, removing the abort button was a good decision and solved a lot of complaints. There are fewer, but still a lot problems with those who leave the game without a move. I think, the automatic abort (which is lesser of two evils) could solve this problems.
The game starts even if one of the players did not make a move and just timed out.
This is not a 3 Player Chess. This is a 4 Player Chess. So, it is critical to have all 4 players at the beginning of the game! By definition.
There is no "Abort" button now, so some players just disconnect or let their flag to fall down. This ruins the game completely. And it must be fixed.
I know that it was proposed many times already. But:
1) I want to emphasize that this is very important.
2) It would be good to hear a word from the game developers on how/when they are going to fix this.