How rare is it for a chess.com user to have balanced stats?

Sure, there are multiple inflationary and deflationary affects.

One funny one is this. If a new player joins, and plays every day for a year... as they improve, they'll lower the ratings of other players.

There are also idiots who, for example, lose every game on purpose (sandbag) or win every game on purpose (cheat) and then get banned.

When two active players play (let's say they both have the same RD) the game doesn't add or subtract any rating points from the pool, so a Hikaru and Naroditsky match doesn't do what tygxc is saying a new GM does. What tygxc is talking about is when the point-to-skill relationship is altered by points being permanently added or removed from the pool.

For example if by some error all your ratings were set to 1000, then those points would disappear, and the cost of returning you to your correct rating would be evenly shared by all other players.

The most likely explanation is as the time duration reduces, flagging becomes more of a dominant factor.  Unfortunately flagging is effected by factors that can have nothing to do with chess skill and are completely out-with the players control.  For example, physiological reaction, local equipment lag, internet lag, and of course if you don't have a paid subscription Ads lag.

On top of this, it may well be that the weaker a player is the less their chess skill can compensation for such factors.

Aside from flagging there may be other non-chess skills that come into play, for example pre-moving. One would expect frequency of play would kick in as well, people are likely to play formats that give the highest rating. and v.v.

Yeah, it’s unlikely. My current stats are about as balanced as they can be, and I’ll quickly tilt in rapid sometime soon anyways.