Firstly, fusion doesn't happen in the way depicted in the question. Four protons don't participate in a 4-body reaction. Instead there are many intermediate steps:
Each step has its own reaction rate. The overall reaction rate is determined by the rate limiting step. The proton-proton reaction is the rate limiting step in this case.
It is important to think in terms of the rate at which a reaction occurs, rather than whether or not it will occur.
The reaction rate will depend on temperature and pressure. In the Sun, pressure is ~265 billon bar, so the reaction can proceed at 10-15 million K. The rate of reaction is actually very low, the center of the Sun only produces energy at rate of ~277 Watts per cubic meter. On Earth, we can not built a reactor with pressure this high, so higher temperature is needed. Different reactions such as starting with deuterium or tritium are used to avoid the need for the proton-proton reaction. Then fusion could be achieved in the ~100 million K range for example.