This is an inaccurate evaluation of the metagame on Pictochat. Even if both players give the stage all the respect it deserves, top level play does not involve both characters vying for control over the "safe zone".
The "safe zone" idea is an egregious oversimplification of the Pictochat dynamic. I only need one point to prove this: very few characters (Pit, Fox, Falco) can apply enough pressure from the "safe zone" for it to be considered overly advantageous to have control over it. Instead, for most matchups, when in neutral positions, controlling the safe zone is only trivially advantageous. This means that there's enough of a difference so that if given the choice, one would obviously choose the safe zone, but in the end, the difference is negligible. In other words, in most matchups, being outside of the safe zone in itself does not put you in any relevant level of danger, because in neutral positions, since there is insufficient additional pressure from your opponent, it is perfectly reasonable to expect the player to be able to dodge hazards on reaction, as hitboxes do not become active until the full drawing has appeared. Because most characters cannot apply enough pressure to take advantage of being in the safe zone, control of the safe zone or lack thereof is not the main motivator for an approach - it is still the case that the one who is pressured to approach is the one who does not have the lead.
In summary, if player A does not have the lead but has control over the safe zone, player B is still not encouraged to approach. Conversely, player A will have to leave the safe zone in order to pressure player B in an attempt to regain the lead. This is generally something he is willing to do, because the motivation from not having the lead trumps the motivation to keep the safe zone. Therefore, the existence of a zone where hazards do not immediately spawn is largely irrelevant to the overarching dynamic of the match. Those characters who are able to apply pressure from the safe zone are also able to apply that pressure from some given spot on Final Destination, so again, it is not control over a safe zone that is forcing approaches.