I do think that there's some merit in sticking with probability on discrete spaces for a while. Once you start dealing with continuous spaces, soon you're talking measure theory and you can wade deep into the technical details and miss some understanding of what's going on. I go back and forth on this as I think it's largely down to the reader to figure out what works for them, but I think probability is one of those fields where developing intuition early on is a must if you want to go further.
The actual requirement for measure theory is overblown. As long as you've taken single and multivariable calculus, you can study continuous probability without any problems and without even knowing what measure theory is.
Agreed, not knowing measure theory never stopped me from computing a conditional expectation. Some courses and books overemphasize rigor in probability and, while it obviously has its place, I've seen newcomers to the field become obsessed with doing everything via measure theory. Further to your point, volume two of Feller is pretty light on measure theory IIRC.
