The rhythm thing is the part everyone fights at first. You're probably letting go of the iron before you trigger, and the joint is already cooling. Keep the iron on the joint the whole time, press the sucker tip right up against the molten blob, then trigger while the iron is still sitting there. Two hands, or if your sucker is the trigger type, set it upright and squeeze the trigger with your thumb on the same hand holding the board. Some people heat the joint, add a tiny bit of fresh solder on top so the iron has something to grip, and that also gives the sucker more molten material to grab. That helps a lot with stubborn joints. Angle matters less than contact. Tip touching the joint, not hovering above it. If it's hovering or bouncing when you trigger, you get splatter. Splatter also happens when the solder isn't fully liquid yet, so give it an extra second than feels neccessary.
The pin coming up with the solder usually means the joint wasn't hot enough, so the whole thing moves as one cold lump. Or you're pushing down too hard with the sucker tip and shoving the lead through. Light touch. If a pin stays stuck, reheat and try again rather than forcing it. The black gunk on the tip is totally normal, that's oxidized solder and flux. Wipe it on a damp sponge or a brass coil like you would an iron tip. If the hole gets clogged, unscrew the nozzle and poke it out now and then. The back side heating trick works but you have to be fast, like under two seconds, or you just feed solder through the hole from the other side. Honestly for a couple of capacitors, desoldering braid with some flux is often easier than a pump for the first joint anyway.
One thing since it's an old stereo: the big electrolytic caps can hold a charge long after it's unplugged. Discharge them before you go yanking on anything.