The problem seems to be thermal resistance not power of heat source. Often professional techs only need a 25W iron with a dimmer set to 60% with a well tinned clean tip. Rule of thumb, if it is not clean or shiny, then the surface is insulated with an oxide coating.
If you have mechanical screws for a gun or threads between the heater and tip, make sure they are clean and tight to reduce the thermal resistance. Then you can reach the 550~600'F required.
FWIW, stained glass leaded wire soldering irons are not hotter, but have more thermal mass and low thermal resistance when properly maintained so they transfer heat efficiently.
Once these steps are understood resin core flux can solder any leaded part in < 3 sec. SMT may take longer and preheat the board is often recommended on a hot plate for rework or a vacuum solder station. Thermal controlled tips work better because they can compensate to some extent for variable thermal resistance in the tip from oxide coating.
The very best irons use RF to heat the surface metal of the tip, rather than a heating element so the response time is <0.1 sec. Of course, they are more expensive and only commercially used. They employ the "Peltier Effect" to regulate temperature in the metal clad coating and can desolder a flatpack SMD IC at the rate of 2~3 IC's per second with special adapters and skill.
So my rule of thumb is, if does not transfer heat enough to solder in < 3 seconds, your tool is oxidized or has loose fittings. Expect ~ 1~3 seconds for a well-maintained iron, to stay as a good as new. This maintenance is before and after every use to keep it tinned to prevent corrosion (oxide). A moist sponge may be used to remove excess solder and oxide. Abrasives may remove the special plating on the tip so be careful and try not to inhale the solder & flux fumes using a fan or vacuum exhaust system for daily work.
You are correct to add solder to pre-heat the joint, but this is due to poor maintenance and can work in a pinch, but then move the solder to the other side to all flow throughout the centre of strands or to wick up the edge.
FYI "60/40: melts between 183–190 °C (361–374 °F)" Contamination from oxides will raise this temperature significantly.
1/16" dia. tip is standard and you may consider larger or smaller depending on the size of the solder pad.
The velocity of heat transfer depends on the metal, mass and heat sink effect if a ground or power plane is attached to the pad. But for steel wire LED's resistors with flash tin plating have a thermal velocity of about 2mm / second, so keep that in mind when soldering.