Your analysis isn't wrong, but the problem is that the end result of Dem infighting is usually ceding control to the GOP. And Republicans, despite their differences, are way better at rallying together and pushing things through whether their base reps support it or not.
The only reason we know names like Collins and Murkowski is because they voice objections all of the time when Republicans put forth bills and appointees and the like. But 90 some-odd percent of the time, no matter how much they're uncertain, they usually end up voting right along with the rest of the pack.
I think one of the main issues with the minimum wage hike battle is that this is the one progressive idea (if you can even call it progressive) that is supported by a fair majority of the people. If you can't get a popular concept like raising the minimum wage through the gates, you have little to no shot at something much less popular or controversial. Progressives are putting all of their eggs in this basket because it's the best basket they have. And centrist Dems are content with tipping it over regardless.