Generating a random probability vector
An exponential distribution $f_U(u)= \lambda \, e^{-\lambda \,u}$, $u \ge 0$, with arbitrary $\lambda>0$ should work. The joint probability
$$f_{\bf u}({\bf u})=\lambda^d \exp[-\lambda (u_1+u_2 +\cdots +u_d)]$$
is constant over the unit simplex. And it's also uniform over any simplex $u_1+u_2 +\cdots +u_d = t$. And the operation $(p_i) := (\frac{U_i}{\sum_{j=1}^d U_j})$ amounts to a radial projection over the unit simplex, which, by geometric similarity, preserves the uniformity.