I had one of those for maybe ten years and it was superb. Loved it. I would buy another in a heartbeat. It completely rocked on small, solid state stuff and maybe would have taken a little more time on a tube socket pin but it always seemed capable of throwing more heat into a joint if needed. OK, wouldn't do a Fender-amp chassis ground connection, but you would not want that much heat for everyday use anyway. I seriously abused it by carrying it around in a cardboard box when I moved from bench to field and eventually mechanically destroyed it and I'm sorry I did. I have seen these things inadvertently left on for days at a time by careless bench techs and still work fine. The tips should last...a hell of a long time. When I first got mine I used the daylights out of it and I still think the tip lasted a couple of years.
It should melt thin, .03x or .06x solder almost instantly, I mean, right now.
Yes, try the 800 degree tips if you are having a problem.
Ditch the lead-free solder and get some real, fresh 60/40 or 63/37 solder.
Keep the tip clean, with sponge, as Sluckey suggested. DO NOT use any sort of abrasive on the tip. That *could* be your problem but you'd really have to file it and remove material to have it not take solder. That comment on your part is a red flag to me. If you have done that...perhaps you've exposed an underlying metal alloy that is not only unfriendly to solder, but may have formed an oxide layer. If so...that could be it. Oxides are terrible heat conductors and they cannot take part in the chemical reaction (yes, there is a chemical reaction, not just heat) that is a valid solder joint.
WHEN YOU SHUT IT OFF you should "put it to bed" by running it through the wetted sponge after powering it down enough to cool it down quite a bit (not just the tip, but the shank, as well) and melt some solder over the tip (it should be noticeably slower than when in normal use) and leave a film, even a blob (the blob will form because the tip has cooled down a lot) and put it back in the stand with that blob on there.