In addition to corrosion issues (sticking calipers) brake fluid with moisture in it will also have a lowered boiling point. The older and wetter the fluid gets, the easier it is to boil. And if you heat the brakes up and boil the fluid, you don't have any braking in that circuit anymore.
DOT 5 fluid is synthetic/silicone fluid. Water does not mix with it. This is good in some respects, the fluid doesn't get saturated over time. But at the same time, any water that gets into the system will bead up and migrate to low spots in the system, where it might or might not flush out with a fluid change. It is BETTER at handling moisture, it's just worth noting that it does it differently, and not in a completely perfect manner.
DOT 5.1 is confusingly named, but it's basically 'super' DOT 4 fluid, it's compatible with DOT 4, no flush needed.
As for removing fluid by bleeding at the calipers/drums - you need to flush until clean fluid comes out. The moisture is dissolved in the normal style of fluid. You won't get droplets of water out, just old dirty (sometimes) looking fluid. Keep bleeding until the fresh clean fluid arrives at that end of the brake system. And, lol, don't run the reservoir dry doing it.
