For the initial steps of nuclear fusion you need to get two protons together and then turn one into a neutron via the weak force interaction.
Both things are important to the overall rate. The weak flavour change of an up quark to a down quark in a proton is a hindrance to the process and means that you have to get a large population of unstable diprotons together in order to create a stable deuteron. i.e. Making the di-proton is easier than getting the di-proton to change into a bound deuteron, so in that sense the weak interaction controls the overall timescale at a given temperature.
Why is forming the deuteron a bottleneck? Well, it's a weak interaction responsible for changing a proton to a neutron, much weaker than the strong nuclear force responsible for allowing the two protons to get near each other in the first place.
It is the hindrance of the weak interaction step that requires the interior temperature of the Sun to be so high. If it were possible for two protons to fuse to a stable state, then much lower temperatures would be required. This is demonstrated by the order of magnitude lower temperatures at which deuterium or lithium fuse with protons, neither of which require any weak interactions but have the same or higher Coulomb barrier between the reactants.
It is an interesting hypothetical quation to ask what would happen if the weak interaction was much stronger, such that a deuteron was always formed when two protons were brought close together? In such a universe I think stars would have much lower internal temperatures, be larger for a given mass (since $T \propto M/R$ from the virial theorem) and much more luminous and short-lived.