Do the products in exothermic reactions have higher kinetic energy than the products? Do they also have lower chemical energy?
In line with the comment of orthocresol, these potential energy diagrams only display the change in the chemical potential as a function of the reaction coordinate.

To clarify: Let's consider the theoretical situation of a reaction mixture in complete vacuum that undergoes a reaction towards a product state. If we assume that the reaction is exothermic and there is no loss of energy due to irradiation, then this means that the products are heated up by an amount of energy that is gained from the loss in chemical potential (i.e. the exothermic reaction). In other words, if you would plot the chemical potential + the kinetic energy (the two terms combined) as a function of the reaction coordinate, you would obtain a straight line due to the law of conservation of energy, because chemical energy is converted to kinetic energy (heat).