Your reasoning is correct. When we say a reaction occurs only with sufficient energy (activation energy), it is truly a tail of a single (pair of) molecule. But the ensemble of molecules still provides molecules of high energy, able to react, and molecules of low energy which will collide and not react. That effect is expressed macroscopically by the reaction kinetics, with a higher rate of collisions resulting in a higher reaction rate. The kinetic effect of temperature is thus continuous and there is no single temperature at which a system has the activation energy to react (as some people believe). It is indeed true that a reaction may become spontaneous in a given direction only above a certain temperature, but that is a matter of thermodynamics and not kinetics.
Despite all of this, it sort of happens that indeed some reactions do seem to begin to take place at a certain temperature while not changing its thermodynamic spontaneity. That is because the kinetic effect of temperature is not linear, it is actually much more intense that a linear model might predict, sometimes doubling the reaction rate at every 10⁰C increase - this is captured by the Arrhenius equation. Sometimes you do have some molecules reacting, but it is so slow at a the temperature you're looking it's not even noticeable. Bring the temperature up and you will see a rapid increase in reaction rate in a short temperature interval. So it looks like it was an on/off transformation.
More often than not in chemistry, the theory predict really small quantities, but the theory is also naive to assume the molecular distribution is continuous and not made of discrete entities as we know. So it sometimes happens that the idea that there no molecules reacting is in this situations actually closer to reality than the theory suggests.
A remark: molecular energy distribution is not normal generally, but follows something like a Maxwell-Boltzmann distribution.