My old iron tip can hold a copious amount of molten solder on it and have a hell lot of thermal mass, so the procedure of soldering SMT components are usually let the tip hold some solder, place the component on the board, touch the joint with the tip of the iron.
What you're describing is a type of drag soldering. The success of this type of soldering is highly dependent on tip shape. Some tips shapes are meant to hold solder (chisel/bevel), and some aren't.
A fine conical tip is the worst kind for this, since it will tend to suck solder away from the tip, up the shaft. With a blunter conical, this technique will work great. 0.2-0.5mm tips are my preference for conicals, for general board-level use.
Also if I accidentally got too much solder on a joint I can just scoop away the excess solder using that big heavy tip.
It sounds to me like you continually created your own tinned-face-only tip with your cheap iron, by applying solder to one spot and letting the rest of the tip oxidize.
If you like to drag solder, I recommend (besides using flux), see if you can find a bevel tip with a tinned face, only. This is denoted by "CF" in the part number. When you use this kind of tip, it holds solder right on the tip, where you want it. And paradoxically, it will suck excess solder out of the joint. So it applies solder, copiously, but leaves only a small amount. In fact,
if you slightly turn the beveled face away from the joint, it will magically remove any bridges left from a reflow oven, etc. Because the entire tinned surface is a flat disc, the surface tension of the molten solder acts like a directional vacuum that melts solder at the edges of the disc and pulls it towards the center (having the beveled tip keeps the surface roughly flat to the board, so this action doesn't have to work against gravity). With a tip like this on your iron, you can throw away your solder wick and desoldering pump. The downside of a CF tip is that it's directional and you can't feed solder onto the top/side while doing lots of solder-hungry connections. You have to lay your solder down on your bench and melt from the top, or you have to turn the tip to apply solder. It's sort of a dedicated tip. For lots of point to point soldering and thru hole parts where you're burning through spools of solder, it's more cumbersome that it's worth. For SMD, only, it can get most any job done, efficiently, with a small learning curve.
Some say a hollow core tip or a spoon tip are the key for drag soldering (CM). They don't make a CM for my iron, so I can't compare. The CF, tinned face-only, makes this technique foolproof.