We are not introducing the hydrogen atom and hydrogen-like ions by means of the Bohr model because it is, after all, a failed model. While the Bohr model correctly incorporates the quantum nature of electronic states (and gets energetics correct, both absolute state energies and excitation energies), it does not satisfy the Heisenberg uncertainty principle. As such, using it in an introductory class may very well encourage students to cling to one of the worst and most intransigent misconceptions they frequently hold about the subject, that electrons orbit nuclei the way that planets orbit the sun, in well-defined paths.

A better place to describe the Bohr model is in an advanced course on quantum chemistry where the historical development is treated after first reviewing the correct, modern formulation.

Why does the Bohr model continue to be used? Granted, it is mathematically simple and correct (as a non-relativistic treatment). Beyond that: in my opinion the format of covering the historical development of quantum mechanics or quantum chemistry may be a holdover from the early years of the field, where it was constantly necessary to persuade and convince people that quantum theory is valid. A big part of "quantum apologetics" is showing the manner in which scientists deduced the characteristics of quantum behavior one experiment at a time, with the concomitant theoretical developments that were fashioned to accommodate the experimental data. It should no longer be necessary to convince most people of quantum realities!

Bohr's theory was groundbreaking when it was proposed, but it is, ultimately, a compromise between classical and quantum mechanics.