Bayesian Optimization is a popular approach for optimizing expensive black-box functions. Its key idea is to use a surrogate model to approximate the objective and, importantly, quantify the associated uncertainty that allows a sequential search of query points that balance exploitation-exploration. Gaussian process (GP) has been a primary candidate for the surrogate model, thanks to its Bayesian-principled uncertainty quantification power and modeling flexibility. However, its challenges have also spurred an array of alternatives whose convergence properties could be more opaque. Motivated by these, we study in this paper an axiomatic framework that elicits the minimal requirements to guarantee black-box optimization convergence that could apply beyond GP-based methods. Moreover, we leverage the design freedom in our framework, which we call Pseudo-Bayesian Optimization, to construct empirically superior algorithms. In particular, we show how using simple local regression, and a suitable "randomized prior" construction to quantify uncertainty, not only guarantees convergence but also consistently outperforms state-of-the-art benchmarks in examples ranging from high-dimensional synthetic experiments to realistic hyperparameter tuning and robotic applications.