This sounds more like a pedagogical issue than an inherent problem with classical statistics. I agree that Bayesianism is philosophically “cleaner”, but classical statistics isn’t just a hodge-podge of unjustified tools. If you’re interested in a sophisticated justification of classical methods, this is a good place to start. I’m pretty sure you’ll be unconvinced, but it should at least give you some idea of where frequentists are coming from.
This sounds more like a pedagogical issue than an inherent problem with classical statistics. I agree that Bayesianism is philosophically “cleaner”, but classical statistics isn’t just a hodge-podge of unjustified tools. If you’re interested in a sophisticated justification of classical methods, this is a good place to start. I’m pretty sure you’ll be unconvinced, but it should at least give you some idea of where frequentists are coming from.