First of all, the lynch is the ONLY thing that is guaranteed to be in the hands of a town. Barring multiball/setup shenanigans, there are always more town than scum in an active game, and town players will have more votes than scum players. So as a whole, the town controls a lynch.
In contrast, night play generally favors scum. The scum get to nightkill. There are often foils to protown power roles (Mafia Roleblockers, Godfathers, etc.). Even protown abilities can wind up hurting the town (like when a vigilante shoots four protown players in four nights).
Giving up your lynches literally hands extra nightkills to the scums.
By extension, No Lynching on
Day One
means:1) Your first night of actions is a series of completely blind actions. This is BAD. Playing a role correctly is a talent, and you give away your ability to play your role correctly for an ENTIRE Day/Night cycle.
2) Your D2 lynch will be uneducated. Yes, there will be results from a scum nightkill and possibly an investigative result. But unless the investigative result is claimed, nobody will know anything, so they're still playing blindly. A D2 lynch after a D1 No-Lynch is almost as random and inaccurate as a D1 lynch.
3) As mentioned before, you're handing a living player to the scums. A town should seek to use every advantage it possibly can, and going into D2 with no bandwagons and a dead protown player is just bad strategy.
4) If a moderator designs a setup for a Day Start, you should assume that the setup is balanced for a Day Start. While assuming moderator competence is not always safe (Blood Bowl Mafia, anyone?), it's a good baseline. Letting the game "start" with night unbalances the setup, and probably NOT in your favor.