You are exactly right. If all molecules in a mixture have enough energy to react, then they all react. If the energy available for the molecules to react is far higher than the energy necessary for either pathway, then you do not see selectivity between the pathways. The reaction will be very fast, and you will see a mixture of products.
This is one reason why you control the temperature in a reaction. The idea is to give enough energy for the reaction pathway that you desire to take place at a reasonable rate, without giving so much energy that other, higher energy pathways become available.
One example where this is sometimes critical is in SN2 reactions. If you have two different functional groups that could both act as nucleophiles present on a molecule, it can be tricky getting the reaction selective for only one of the two functional groups to add. However, the stronger nucleophile will react faster (that reaction has a lower activation energy), and at lower temperatures you may see selectivity while at higher temperatures you see none.
This is also useful in selecting between the kinetic and the thermodynamic products of a reaction, but that is a whole new lecture.