The Sun is cooler than the proton barrier suggests. Why does fusion still start?

Answer: Quantum tunnelling tail

Classical high heat — Classical high heat is the obvious guess: make protons fast enough and they should climb the electric barrier. Early theorists noticed the Sun's core was too cool for that simple picture. The Sun's temperature matters, but tunnelling supplies the extra chance that classical collisions lack.

Quantum tunnelling tail ✓ — Correct. Two protons repel electrically, but their quantum wavefunctions give a small chance of penetrating the Coulomb barrier and getting close enough for the strong interaction. The rate for one pair is tiny; the Sun works because it contains an enormous number of protons. Starlight depends on rare failures of classical repulsion.

Pressure-only squeeze — Pressure does help the Sun by packing hot protons into a dense core, so the guess is close to the everyday picture of squeezing things together. But pressure alone would still leave the electric barrier too high for ordinary collisions at solar-core temperatures. The missing piece is the quantum tail that gives rare close approaches.