Tinned face-only bevel tips changed everything for me. I don't know if Weller has the same thing. For my hakko, it is T18-CF3. 3mm bevel tip with chrome jacket over the sides so only the face can hold solder. (I have them in all sizes, but 3mm is best for drag soldering and bridge removal, IMO).
On a shoulder bridge, you touch the iron to the top of the pins by the chip and slide the tip down off the end of the pins. It sucks like a hoover. But unlike a conical, chisel, or regular bevel, the solder doesn't wick up the side of the iron where it must be wiped off. It forms a thin meniscus on the bottom of the iron, so you can put it back on the board where it is supposed to go. This sounds counterintuitive, but it works like black magic. Some kind of surface tension and gravity thing going that pulls/keeps most of the solder on the cut face of the iron and gives/leaves just about the right amount on the pads... as long as you have roughly the right amount of solder on there to begin with. This is why the 3mm works best... larger surface area makes it more forgiving and able to deal with larger amounts of solder; but the 4mm is too big (too large to maneuver for one thing. But it is also too large to hold a huge drop of solder... the surface tension isn't strong enough to hold the drop that forms when you get larger than the 3mm; and this ability is useful for other things like hand soldering of passives and SOT23 and the like; so while it can hold a wee bit more solder than the 3mm, it is easy to add too much and drop the bead, and it is not worth the decrease in maneuverability.)
Other major benefit is the entire wettable surface can be kept with a layer of solder most of the time under use, on the bottom face where gravity naturally pulls melted flux. So the tip rarely needs cleaning. If you keep things moving, you don't have to stop to clean the tip, ever.
The nonwettable tips sounds like they might be good for removing bridges post oven-soldering where the amount of solder is very small; where it doesn't have to be removed, just moved. If you're hand soldering to begin with, it is just an extra step...