  We develop an improved version of the stochastic semigroup approach to study
the edge of $\beta$-ensembles pioneered by Gorin and Shkolnikov, and later
extended to rank-one additive perturbations by the author and Shkolnikov. Our
method is applicable to a significantly more general class of random
tridiagonal matrices than that considered in these previous works, including
some non-symmetric cases that are not covered by the stochastic operator
formalism of Bloemendal, Ram\'irez, Rider, and Vir\'ag.
  We present two applications of our main results: Firstly, we prove the
convergence of $\beta$-Laguerre-type (i.e., sample covariance) random
tridiagonal matrices to the stochastic Airy semigroup and its rank-one spiked
version. Secondly, we prove the convergence of the eigenvalues of a certain
class of non-symmetric random tridiagonal matrices to the spectrum of a
continuum Schr\"odinger operator with Gaussian white noise potential.
